Properties of Exponents - Algebra 2

00:59:14
https://www.youtube.com/watch?v=etMK3xViMAc

概要

TLDRThe video thoroughly covers exponent rules including multiplication and division of exponents with the same base, methods for handling negative exponents, and how to raise an exponent to another exponent. It explains the importance of positive exponents and provides practical examples for learners to follow along. Practice problems are included to reinforce understanding and ensure application of learned concepts. The relationship between exponent rules and roots is also explored, aiding in comprehension of radical expressions.

収穫

  • ➕ When multiplying, add the exponents.
  • ➖ When dividing, subtract the exponents.
  • 🔄 Negative exponents can be made positive by flipping the base.
  • 0 Exponent means the value is 1.
  • ✖️ Raising an exponent to another power means multiplying the exponents.
  • 📚 Practice with examples to solidify understanding.
  • ➕ Multiple variables' exponents are added separately for each variable.
  • 📏 Radicals relate directly to fractional exponents.

タイムライン

  • 00:00:00 - 00:05:00

    The video introduces exponents, starting with basic rules for multiplying variables with the same base, specifically explaining how to add exponents when multiplying like terms.

  • 00:05:00 - 00:10:00

    It discusses division of variables with the same base, illustrating that exponents can be subtracted in such cases and how negative exponents can be converted to positive by flipping the fraction.

  • 00:10:00 - 00:15:00

    Examples are provided for multiplying and dividing variables with exponents, guiding viewers through the process of adding and subtracting exponents to arrive at the simplified forms.

  • 00:15:00 - 00:20:00

    The video continues with practice problems for multiplication and divisions, demonstrating how to handle negative exponents and ensure final answers are presented without them.

  • 00:20:00 - 00:25:00

    Different techniques for approaching problems with negative exponents are shared, elaborating on ways to methodically arrive at solutions through strategic simplification.

  • 00:25:00 - 00:30:00

    The video then transitions to discussing the power of an exponent raised to another exponent, emphasizing that in such cases the exponents should be multiplied.

  • 00:30:00 - 00:35:00

    It presents several problems to illustrate the process and reinforces the need for careful attention to the distribution of exponents over products and sums.

  • 00:35:00 - 00:40:00

    Following this, the video integrates basic operations with exponents, like addition and multiplication, and explains how the same bases allow for addition of exponents.

  • 00:40:00 - 00:45:00

    The viewer is encouraged to practice applying these rules, especially in complex expressions with multiple bases and exponents.

  • 00:45:00 - 00:50:00

    Radicals are introduced, focusing on converting expressions in exponential form back to radical form, and demonstrating how to simplify these expressions effectively while retaining clarity on exponents.

  • 00:50:00 - 00:59:14

    Lastly, the video summarizes key concepts related to exponents and radicals, encouraging viewers to practice and apply the learned rules for comprehension.

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ビデオQ&A

  • What happens when multiplying exponents with the same base?

    You add the exponents together.

  • What happens when dividing exponents with the same base?

    You subtract the exponents.

  • How do you handle negative exponents?

    You convert them to positive by flipping the base to the denominator.

  • What is any exponent raised to the zero power?

    It equals 1.

  • How do you raise an exponent to another exponent?

    You multiply the two exponents together.

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  • 00:00:01
    so this video will give you an
  • 00:00:02
    introduction to exponents
  • 00:00:05
    but let's begin let's go over some basic
  • 00:00:07
    rules
  • 00:00:08
    so what happens when
  • 00:00:10
    you multiply
  • 00:00:12
    two numbers with the same variable
  • 00:00:15
    what do you do
  • 00:00:17
    x cubed times x to the fifth power
  • 00:00:21
    is x to the eighth power you simply
  • 00:00:24
    add the exponents
  • 00:00:26
    but now why is that
  • 00:00:30
    let's use a simpler example x squared
  • 00:00:32
    times x cubed
  • 00:00:33
    two plus three is five
  • 00:00:36
    x squared
  • 00:00:37
    means you're multiplying two x variables
  • 00:00:39
    together
  • 00:00:40
    that's x squared
  • 00:00:43
    now x cubed
  • 00:00:45
    means that you're multiplying three x
  • 00:00:48
    variables together
  • 00:00:49
    so what you're really doing if you
  • 00:00:50
    combine them you're multiplying a total
  • 00:00:52
    of five x variables together which is x
  • 00:00:55
    to the fifth power and that's what it
  • 00:00:57
    means
  • 00:00:59
    so whenever you multiply common
  • 00:01:00
    variables
  • 00:01:01
    you're allowed to add the exponents
  • 00:01:05
    so now what if we're dividing
  • 00:01:08
    by two common variables what can we do
  • 00:01:10
    now if you're dividing you can subtract
  • 00:01:13
    8 minus 3 is 5.
  • 00:01:16
    so now what about like
  • 00:01:18
    x to the third divided by x to the fifth
  • 00:01:20
    power
  • 00:01:24
    x to the third divided by x to the fifth
  • 00:01:26
    you can subtract three minus five
  • 00:01:29
    three minus five is negative two
  • 00:01:32
    now if you get a negative exponent you
  • 00:01:34
    can make it positive
  • 00:01:36
    by flipping the fraction if the x is on
  • 00:01:38
    top you can move it to the bottom if you
  • 00:01:40
    do that
  • 00:01:41
    it's going to change to positive 2.
  • 00:01:44
    now let's see
  • 00:01:45
    another way in which we can look at the
  • 00:01:46
    situation
  • 00:01:48
    now we said that x cubed is basically
  • 00:01:51
    three x variables multiplied to each
  • 00:01:53
    other and x to the fifth power
  • 00:01:55
    is five x variables
  • 00:01:57
    multiplied to each other
  • 00:01:59
    so we can cancel these two
  • 00:02:02
    we could cancel another x
  • 00:02:04
    and another one
  • 00:02:07
    so notice that we have no x variables
  • 00:02:09
    left on top x divided by x is one
  • 00:02:12
    on the bottom
  • 00:02:13
    we have two x variables left over x
  • 00:02:16
    times x is x squared
  • 00:02:18
    so that's another way in which you can
  • 00:02:19
    get the same answer
  • 00:02:21
    or if you subtract it backwards five
  • 00:02:24
    minus three is equal to two
  • 00:02:30
    now let's put some of these rules that
  • 00:02:32
    you've just learned into practice
  • 00:02:34
    so go ahead and multiply
  • 00:02:37
    these variables
  • 00:02:57
    feel free to pause the video
  • 00:03:00
    and work out these
  • 00:03:02
    examples
  • 00:03:04
    so x to the fourth times x to the fifth
  • 00:03:07
    we know we need to add the exponents
  • 00:03:10
    four plus five
  • 00:03:12
    is equal to nine
  • 00:03:14
    so that's it for the first one
  • 00:03:16
    now for the second one let's add
  • 00:03:18
    what is seven plus negative three
  • 00:03:22
    seven
  • 00:03:22
    plus negative three is the same as seven
  • 00:03:25
    minus three
  • 00:03:26
    which is four
  • 00:03:29
    for the next one
  • 00:03:33
    it's going to be 4
  • 00:03:34
    plus negative 6 which is 4 minus 6.
  • 00:03:38
    4 minus 6 is negative 2.
  • 00:03:41
    and because we have a negative exponent
  • 00:03:42
    we need to simplify and rewrite it as 1
  • 00:03:45
    over x squared
  • 00:03:49
    you never want to leave your final
  • 00:03:50
    answer as a negative exponent you want
  • 00:03:51
    to make it positive
  • 00:03:54
    so if the x variable is on top move it
  • 00:03:56
    to the bottom
  • 00:03:58
    here 3 plus negative eight or three
  • 00:04:00
    minus eight is negative five
  • 00:04:03
    which is one over x to the fifth power
  • 00:04:09
    now let's practice some problems on
  • 00:04:11
    dividing variables let's say
  • 00:04:14
    try these if you have x is 7
  • 00:04:16
    divided by x to the fourth
  • 00:04:20
    x to the 5 over x to the minus 2
  • 00:04:26
    x to the negative 3 over x to the sixth
  • 00:04:29
    power
  • 00:04:30
    and
  • 00:04:32
    this one as well
  • 00:04:36
    so x is seven divided by x to the four
  • 00:04:38
    that's going to be seven minus four
  • 00:04:40
    which is three
  • 00:04:42
    now for the next one
  • 00:04:44
    we need to subtract the top one five
  • 00:04:47
    minus the bottom number so five minus
  • 00:04:49
    negative two
  • 00:04:51
    five minus negative two is the same as
  • 00:04:53
    five plus two whenever you have two
  • 00:04:55
    negative signs next to each other you
  • 00:04:57
    can make it positive
  • 00:04:58
    so the final answer is x to the seventh
  • 00:05:00
    power
  • 00:05:03
    now for this one it's going to be the
  • 00:05:05
    top exponent negative three
  • 00:05:08
    minus the
  • 00:05:09
    exponent on the bottom six
  • 00:05:11
    negative three minus six is negative
  • 00:05:13
    nine so this is x
  • 00:05:15
    to the negative nine
  • 00:05:16
    since we have a negative exponent we
  • 00:05:18
    need to fix it
  • 00:05:19
    and that's gonna become x to the nine
  • 00:05:23
    now another way in which you could see
  • 00:05:24
    this problem you can do it this way
  • 00:05:27
    you can take this variable
  • 00:05:29
    and move it to the bottom if you do so
  • 00:05:31
    the negative exponent will become
  • 00:05:32
    positive
  • 00:05:34
    so you can write it as x cubed
  • 00:05:36
    and x to the six on the bottom and we
  • 00:05:38
    know x to the third times x to the sixth
  • 00:05:41
    is three plus six which is nine
  • 00:05:43
    so you get the same answer one over x to
  • 00:05:45
    the nine
  • 00:05:46
    so as you can see there's multiple ways
  • 00:05:48
    in which you can get the answer you have
  • 00:05:50
    to find a method that is convenient for
  • 00:05:51
    you
  • 00:05:54
    now this one is definitely some
  • 00:05:55
    different techniques that we can use
  • 00:05:56
    here
  • 00:05:57
    let's make some space first
  • 00:06:09
    now we need to subtract the top exponent
  • 00:06:12
    negative five
  • 00:06:13
    minus
  • 00:06:15
    the exponent on the bottom negative four
  • 00:06:17
    so this is
  • 00:06:19
    negative five plus four
  • 00:06:23
    which is uh
  • 00:06:24
    negative one
  • 00:06:26
    and that's the same as one divided by x
  • 00:06:28
    to the first power or simply one over x
  • 00:06:33
    now we can move the x to the negative 5
  • 00:06:35
    to the bottom to make it positive
  • 00:06:37
    or we could take the other variable move
  • 00:06:39
    it to the top to make it positive as
  • 00:06:41
    well or we could just flip the whole
  • 00:06:42
    fraction
  • 00:06:43
    if we do that
  • 00:06:45
    it's going to look like this
  • 00:06:50
    now if we subtract it forwards like top
  • 00:06:53
    minus bottom
  • 00:06:54
    this is going to be x to the negative
  • 00:06:56
    one
  • 00:06:57
    if you subtract it backwards bottom line
  • 00:06:59
    is the top number five minus four you're
  • 00:07:02
    gonna get x to the one on the bottom
  • 00:07:07
    by the case you get the same thing four
  • 00:07:08
    minus five is negative one and you can
  • 00:07:10
    flip it you get this
  • 00:07:13
    let's try another example like that
  • 00:07:15
    so if you have x to the
  • 00:07:18
    negative 4 divided by x
  • 00:07:21
    to the negative 7
  • 00:07:23
    you can flip it
  • 00:07:25
    and then it's simply 7 minus 4 which is
  • 00:07:27
    3.
  • 00:07:31
    so let's say
  • 00:07:33
    if we have this example
  • 00:07:35
    this is going to be x to the third
  • 00:07:37
    divided by x to the eighth
  • 00:07:40
    and then
  • 00:07:41
    three minus eight is negative five which
  • 00:07:44
    is one over x to the fifth power
  • 00:07:53
    now what if you were to see something
  • 00:07:54
    like this one over x to the negative
  • 00:07:56
    four
  • 00:07:58
    what's the answer
  • 00:08:00
    well this you could just simply move it
  • 00:08:01
    to the top
  • 00:08:03
    and it's x to the fourth power
  • 00:08:06
    now what about
  • 00:08:07
    x to the zero power what's the answer
  • 00:08:11
    anything raised to the 0 power is 1.
  • 00:08:15
    so what if you have 3 x to the 0 power
  • 00:08:18
    what's the answer now
  • 00:08:19
    as opposed to 3 x
  • 00:08:22
    to the zero
  • 00:08:25
    in the first example the zero only
  • 00:08:27
    applies to the x so this is going to be
  • 00:08:29
    three times one which is three
  • 00:08:32
    in the second example the zero applies
  • 00:08:34
    to the three any x so that's three to
  • 00:08:36
    the zero x to the zero which is
  • 00:08:37
    basically
  • 00:08:39
    just one
  • 00:08:40
    the whole thing is one
  • 00:08:43
    so knowing that
  • 00:08:44
    try these
  • 00:09:06
    so x y to the zero
  • 00:09:08
    is one but we have a negative four
  • 00:09:10
    outside of it so the whole thing is
  • 00:09:12
    equal to negative four
  • 00:09:14
    here the zero applies to everything in
  • 00:09:16
    this problem so
  • 00:09:17
    the entire answer is one
  • 00:09:20
    and for this it's 5 minus 34
  • 00:09:23
    and a 0 only applies to this portion
  • 00:09:25
    which is equal to 1
  • 00:09:27
    and 5 minus 34
  • 00:09:30
    is negative 29.
  • 00:09:37
    now
  • 00:09:38
    what happens
  • 00:09:40
    if you have multiple variables let's say
  • 00:09:46
    if we have x squared y cubed
  • 00:09:49
    times
  • 00:09:50
    x to the third y to the fourth
  • 00:09:54
    let's go back to multiplying variables
  • 00:09:55
    but now we have multiple examples
  • 00:09:59
    so if you see a problem like this focus
  • 00:10:01
    on
  • 00:10:02
    the x variables first
  • 00:10:06
    and then separately we'll multiply
  • 00:10:08
    the y variables
  • 00:10:10
    so what's x squared times x cubed
  • 00:10:13
    so you have to multiply like terms
  • 00:10:16
    2 plus three is five
  • 00:10:20
    and
  • 00:10:20
    three plus four is seven
  • 00:10:23
    so you could simply leave it as x to the
  • 00:10:25
    fifth y to the seventh
  • 00:10:29
    try this example what
  • 00:10:30
    is two x cubed
  • 00:10:34
    y to the fourth times five
  • 00:10:37
    x to the negative seven
  • 00:10:38
    y to the third
  • 00:10:41
    so first let's multiply two times five
  • 00:10:44
    two times five oh by the way for each
  • 00:10:46
    example
  • 00:10:47
    make sure you um make sure you pause the
  • 00:10:49
    video
  • 00:10:50
    and work out the examples yourself to
  • 00:10:52
    see if you can get the answer and then
  • 00:10:54
    unpause it to see what the answer
  • 00:10:56
    actually is
  • 00:10:58
    two times five is ten
  • 00:11:02
    x to the third times x to the negative
  • 00:11:04
    seven
  • 00:11:05
    we know three plus negative seven
  • 00:11:09
    is equal to negative four so this is x
  • 00:11:13
    raised to the minus four
  • 00:11:15
    and if we add
  • 00:11:17
    four plus three
  • 00:11:19
    that's seven
  • 00:11:20
    now the only thing we need to do is get
  • 00:11:22
    rid of the negative exponent
  • 00:11:24
    so we need to put it to the bottom so
  • 00:11:25
    the final answer is ten y to the seventh
  • 00:11:29
    divided by x to the fourth power
  • 00:11:33
    try this example
  • 00:11:35
    what is three
  • 00:11:36
    x to the negative five
  • 00:11:38
    y to the third
  • 00:11:40
    z to the fourth
  • 00:11:42
    time seven
  • 00:11:44
    x to the second
  • 00:11:47
    y to the negative six
  • 00:11:49
    z to the seventh
  • 00:11:51
    so let's multiply three times seven
  • 00:11:54
    three times seven is 21
  • 00:11:57
    and then
  • 00:11:58
    negative 5 plus 2
  • 00:12:01
    is negative 3
  • 00:12:04
    3 plus negative 6
  • 00:12:07
    is also negative 3.
  • 00:12:14
    and finally
  • 00:12:15
    four plus seven
  • 00:12:18
    is eleven
  • 00:12:19
    now we have two negative exponents let's
  • 00:12:22
    move it to the bottom
  • 00:12:23
    so therefore this is going to be 21
  • 00:12:26
    z to the 11th power
  • 00:12:28
    divided by x cubed y cubed
  • 00:12:35
    now what about division
  • 00:12:37
    so let's say if we have uh
  • 00:12:41
    36
  • 00:12:42
    x to the fifth
  • 00:12:44
    y to the third
  • 00:12:46
    z to the negative fourth
  • 00:12:48
    divided by
  • 00:12:49
    nine
  • 00:12:51
    x squared
  • 00:12:52
    y to the seventh
  • 00:12:55
    and
  • 00:12:56
    z to the eighth
  • 00:12:59
    so first let's divide thirty six by nine
  • 00:13:02
    thirty six divided by nine is four
  • 00:13:05
    next let's divide x to the fifth power
  • 00:13:07
    divide by x squared
  • 00:13:09
    five minus two
  • 00:13:11
    is three
  • 00:13:14
    three minus seven
  • 00:13:17
    is negative four
  • 00:13:20
    and
  • 00:13:21
    negative four minus eight let's write
  • 00:13:22
    that
  • 00:13:23
    negative four minus eight is negative
  • 00:13:25
    twelve
  • 00:13:26
    so this is going to be z
  • 00:13:28
    to the negative twelve
  • 00:13:30
    now let's get rid of the negative
  • 00:13:31
    exponents
  • 00:13:32
    so x cubed will remain on top
  • 00:13:35
    the other variables we need to put it in
  • 00:13:37
    the bottom so that it's going to be
  • 00:13:38
    positive so it's y to the fourth
  • 00:13:41
    and then z to the 12th power
  • 00:13:44
    so this is the answer
  • 00:13:49
    now let's try
  • 00:13:53
    this one let's say if it's 35
  • 00:13:56
    x to the negative four
  • 00:13:59
    y to the negative seven
  • 00:14:01
    z to the eighth power
  • 00:14:03
    divided by 49
  • 00:14:06
    x to the sixth y to the minus three
  • 00:14:09
    and z to the negative four
  • 00:14:13
    so we can't divide 35 by 49 nicely
  • 00:14:17
    it's not like 36 over 9 which gave us a
  • 00:14:19
    whole number however
  • 00:14:21
    35 and 49
  • 00:14:23
    are both divisible by 7
  • 00:14:25
    so we can reduce it
  • 00:14:28
    35 is basically seven times five
  • 00:14:30
    49 is seven times seven
  • 00:14:33
    so we can cancel a seven
  • 00:14:35
    and we will be left with five divided by
  • 00:14:38
    seven
  • 00:14:40
    so let's put a fraction let's put a five
  • 00:14:42
    on top and a 7 on the bottom
  • 00:14:45
    now negative 4 minus 6
  • 00:14:49
    is equal to negative 10.
  • 00:14:52
    so initially everything will be on top
  • 00:14:56
    now negative seven minus three
  • 00:15:00
    minus negative three
  • 00:15:02
    is the same as negative seven plus three
  • 00:15:05
    which is negative four
  • 00:15:06
    so right now we have y to the minus four
  • 00:15:12
    now keep in mind you can use the reverse
  • 00:15:14
    technique method
  • 00:15:16
    you can flip these two
  • 00:15:18
    and say it's
  • 00:15:20
    y to the third divided by y to the
  • 00:15:22
    seventh
  • 00:15:23
    and three minus seven is negative four
  • 00:15:25
    so you'll get the same result
  • 00:15:28
    now eight
  • 00:15:29
    minus negative four
  • 00:15:31
    eight minus negative four is the same as
  • 00:15:33
    eight plus four
  • 00:15:35
    which is equal to twelve
  • 00:15:37
    so we have z to the twelve which will
  • 00:15:39
    remain on top
  • 00:15:41
    the ones with the negative exponents
  • 00:15:42
    we're going to move it to the bottom so
  • 00:15:44
    this is 5 z to the 12th
  • 00:15:47
    divided by 7
  • 00:15:48
    x to the 10th
  • 00:15:50
    y to the fourth
  • 00:15:51
    so this is the answer
  • 00:15:57
    now let's try one more example like that
  • 00:16:00
    so let's say if we have 63
  • 00:16:04
    x to the nine
  • 00:16:05
    y to the negative six
  • 00:16:07
    and z
  • 00:16:09
    to the negative
  • 00:16:10
    10
  • 00:16:11
    divided by
  • 00:16:13
    36
  • 00:16:15
    x to the negative 4
  • 00:16:17
    y to the fifth and z to the negative 7.
  • 00:16:21
    try this
  • 00:16:22
    so let's focus on the number 63 and 36
  • 00:16:30
    63
  • 00:16:32
    is 9 times 7
  • 00:16:35
    36 is nine times four
  • 00:16:38
    so both of these numbers are divisible
  • 00:16:40
    by nine
  • 00:16:41
    if you divide sixty three by nine you
  • 00:16:43
    get seven thirty six divided by nine is
  • 00:16:45
    four
  • 00:16:46
    so we're gonna have a seven on top
  • 00:16:48
    and a four on the bottom
  • 00:16:51
    nine minus negative four
  • 00:16:55
    is the same as nine plus four which is
  • 00:16:57
    thirteen
  • 00:17:03
    negative six minus five
  • 00:17:07
    is equal to negative eleven
  • 00:17:15
    negative ten
  • 00:17:17
    minus
  • 00:17:19
    negative seven
  • 00:17:20
    is the same as negative ten plus seven
  • 00:17:23
    which is negative three
  • 00:17:32
    now the last thing that we need to do
  • 00:17:35
    is take the x the negative exponents and
  • 00:17:37
    move it to the bottom so this is going
  • 00:17:39
    to be 7
  • 00:17:40
    x to the 13
  • 00:17:42
    divided by 4
  • 00:17:44
    y to the 11
  • 00:17:46
    z to the third
  • 00:17:50
    now what if we raise
  • 00:17:52
    one exponent to another exponent
  • 00:17:55
    for example
  • 00:17:56
    what is x to the third power raised to
  • 00:17:58
    the fourth
  • 00:18:00
    whenever you raise one exponent to
  • 00:18:02
    another power
  • 00:18:03
    you can multiply
  • 00:18:05
    this is three times four
  • 00:18:07
    which is twelve
  • 00:18:10
    now
  • 00:18:11
    let's understand why
  • 00:18:14
    x cubed raised to the fourth power
  • 00:18:16
    means that you have four
  • 00:18:18
    x cubes multiplied to each other
  • 00:18:22
    now x cubed times x cubed we need to add
  • 00:18:25
    that's x to the sixth power and if we
  • 00:18:27
    multiply these two
  • 00:18:28
    it's x to the sixth power
  • 00:18:31
    six plus six is twelve
  • 00:18:32
    so we're going to get the same answer
  • 00:18:36
    so whenever you raise one exponent to
  • 00:18:38
    another you are allowed to multiply the
  • 00:18:41
    two exponents
  • 00:18:43
    so knowing that
  • 00:18:45
    try these examples
  • 00:19:02
    four times five is twenty
  • 00:19:06
    three times negative two
  • 00:19:08
    is negative six which is one divided by
  • 00:19:12
    x to the sixth power
  • 00:19:14
    negative four times negative six
  • 00:19:17
    is twenty four
  • 00:19:24
    what about these
  • 00:19:26
    let's say if you have
  • 00:19:27
    x squared
  • 00:19:29
    y cubed
  • 00:19:30
    raised to the fourth what's the answer
  • 00:19:33
    in this case you need to distribute the
  • 00:19:35
    four
  • 00:19:36
    four times two is eight
  • 00:19:40
    four times three
  • 00:19:42
    is twelve
  • 00:19:43
    so that's the answer
  • 00:19:45
    now what if it's uh two
  • 00:19:48
    x cubed
  • 00:19:49
    y to the negative four
  • 00:19:52
    raised to the negative third power or
  • 00:19:54
    positive three
  • 00:19:56
    what is the answer
  • 00:20:00
    so
  • 00:20:01
    first we need to know what exponent is
  • 00:20:04
    attached to the two if there's no number
  • 00:20:07
    there's a one
  • 00:20:09
    three times one
  • 00:20:10
    is three
  • 00:20:12
    three times three
  • 00:20:14
    is nine
  • 00:20:17
    and three times negative four
  • 00:20:20
    is negative twelve now what is two to
  • 00:20:22
    the third power
  • 00:20:24
    two to the third power is basically two
  • 00:20:26
    times two times two you're multiplying
  • 00:20:28
    three twos
  • 00:20:30
    two times two is four four times two is
  • 00:20:32
    eight so 2 cubed is 8.
  • 00:20:36
    so this is going to be 8
  • 00:20:39
    x to the 9th and we need to move this to
  • 00:20:41
    the bottom divided by y
  • 00:20:43
    to the 12th power
  • 00:20:48
    try this one
  • 00:20:52
    for
  • 00:20:53
    x to the negative fourth
  • 00:20:55
    y to the fifth
  • 00:20:57
    raised to the second power
  • 00:21:00
    so this is equal to four squared
  • 00:21:03
    negative four times two
  • 00:21:05
    is negative eight
  • 00:21:08
    and
  • 00:21:09
    two times five is ten
  • 00:21:12
    four squared that's four times four
  • 00:21:13
    which is sixteen
  • 00:21:15
    y is going to remain on top but
  • 00:21:18
    x has a negative exponent so we need to
  • 00:21:20
    move it to the bottom
  • 00:21:22
    so this is the final answer
  • 00:21:27
    try this one three x cubed y to the
  • 00:21:31
    negative fourth
  • 00:21:33
    z to the fifth raised to the negative
  • 00:21:35
    fourth power
  • 00:21:38
    so let's distribute the negative four
  • 00:21:40
    one times negative four
  • 00:21:42
    is negative four
  • 00:21:43
    3 times negative 4
  • 00:21:45
    is negative 12 negative 4 times negative
  • 00:21:48
    4 is positive 16
  • 00:21:50
    and 5 times negative 4 is negative 20.
  • 00:21:55
    so the ones with the negative exponents
  • 00:21:57
    will go on the bottom so y to the 16
  • 00:22:00
    will remain on top
  • 00:22:01
    then we're going to have 3 to the
  • 00:22:03
    positive 4 x to the positive 12
  • 00:22:06
    and z to the positive 20 on the bottom
  • 00:22:09
    now let's focus on the value of 3 to the
  • 00:22:11
    fourth power what is that equal to
  • 00:22:14
    three to the fourth is basically three
  • 00:22:17
    times three times three times three
  • 00:22:19
    now three times three is nine
  • 00:22:21
    and the other two threes multiply to
  • 00:22:23
    give you nine nine times nine is eighty
  • 00:22:25
    one
  • 00:22:27
    so the final answer is y to the
  • 00:22:28
    sixteenth power divided by eighty one
  • 00:22:31
    x to the twelve
  • 00:22:33
    z to the twentieth
  • 00:22:38
    now what about this let's say if it's
  • 00:22:40
    three
  • 00:22:42
    x to the fourth
  • 00:22:43
    y to the negative fifth
  • 00:22:45
    divided by two
  • 00:22:47
    x to the seventh
  • 00:22:49
    y to the third
  • 00:22:51
    and let's say it's raised to the second
  • 00:22:53
    power
  • 00:22:55
    now you can apply the exponent first or
  • 00:22:58
    you can divide first
  • 00:23:01
    let's apply the exponent first
  • 00:23:06
    so 1 times two
  • 00:23:09
    is two
  • 00:23:12
    two times four
  • 00:23:14
    is eight
  • 00:23:17
    two times negative five
  • 00:23:19
    is negative ten
  • 00:23:23
    now two times the one below
  • 00:23:25
    that's gonna be two
  • 00:23:31
    seven times two is fourteen and three
  • 00:23:34
    times two is six so now we can divide
  • 00:23:37
    three squared is nine
  • 00:23:39
    two squared is four two times two is
  • 00:23:41
    four
  • 00:23:43
    and
  • 00:23:44
    eight minus fourteen
  • 00:23:47
    is negative six on top
  • 00:23:49
    but if you subtract it backwards 14
  • 00:23:52
    minus 8 that's positive 6 on the bottom
  • 00:23:55
    so we can just go ahead and put that on
  • 00:23:57
    the bottom
  • 00:23:59
    negative 10 minus 6
  • 00:24:02
    is negative 16 on top
  • 00:24:07
    and the final answer
  • 00:24:09
    is going to be
  • 00:24:11
    9 divided by
  • 00:24:13
    4 x to the 6 y to the 16.
  • 00:24:19
    let's work on another example
  • 00:24:21
    so let's say
  • 00:24:24
    if we have
  • 00:24:27
    four
  • 00:24:29
    x to the seventh
  • 00:24:31
    y to the
  • 00:24:33
    negative three
  • 00:24:34
    divided by three
  • 00:24:37
    x to the negative four
  • 00:24:39
    y to the fifth power
  • 00:24:41
    and we're gonna raise it to the third
  • 00:24:42
    power this time
  • 00:24:46
    so what we're going to have is
  • 00:24:48
    1 times 3 is 3
  • 00:24:50
    7 times 3 is 21
  • 00:24:53
    and 3 times negative 3 is
  • 00:24:56
    negative 9.
  • 00:24:59
    now one times three is three
  • 00:25:07
    negative four
  • 00:25:09
    times three
  • 00:25:10
    that's going to be negative twelve
  • 00:25:12
    and finally five times three is fifteen
  • 00:25:16
    so now what's four cubed what's four
  • 00:25:18
    times four times four
  • 00:25:20
    four times four is sixteen sixteen times
  • 00:25:23
    four is sixty-four
  • 00:25:27
    three times three is nine times another
  • 00:25:29
    three is twenty-seven
  • 00:25:32
    twenty-one minus negative twelve
  • 00:25:37
    that's the same as twenty-one plus
  • 00:25:38
    twelve
  • 00:25:40
    which is thirty-three so we have x to
  • 00:25:43
    the 33 on top
  • 00:25:46
    now negative 9
  • 00:25:48
    minus 15.
  • 00:25:49
    if we subtract these two what will we
  • 00:25:52
    get
  • 00:25:53
    this is negative 24 on top
  • 00:25:57
    so now the final answer
  • 00:26:00
    whenever you subtract two exponents the
  • 00:26:02
    top one minus the bottom one the result
  • 00:26:04
    immediately goes on top
  • 00:26:05
    and then if it's negative you can move
  • 00:26:07
    it to the bottom
  • 00:26:08
    so the final answer is 64
  • 00:26:10
    x to the 33 divided by 27
  • 00:26:14
    y to the 24th
  • 00:26:20
    now let's say if we have
  • 00:26:25
    three
  • 00:26:27
    x to the negative third
  • 00:26:30
    y to the negative second
  • 00:26:32
    over
  • 00:26:33
    let's say
  • 00:26:35
    two
  • 00:26:37
    x to the negative
  • 00:26:38
    seven
  • 00:26:39
    y to the negative five
  • 00:26:42
    let's say it's raised to the negative
  • 00:26:44
    fourth power
  • 00:26:46
    now notice that you have a lot of
  • 00:26:49
    negative exponents on the inside
  • 00:26:54
    and we have this negative fourth power
  • 00:26:57
    what should we do in this case
  • 00:27:02
    one thing you could do if you want to
  • 00:27:05
    is
  • 00:27:06
    you could change the negative 4 into a
  • 00:27:08
    positive 4
  • 00:27:09
    by flipping everything
  • 00:27:12
    so everything on the bottom is going to
  • 00:27:14
    move to the top
  • 00:27:17
    but the signs will remain the same
  • 00:27:25
    because
  • 00:27:26
    we're changing only this sign
  • 00:27:28
    so now it's positive 4.
  • 00:27:32
    now
  • 00:27:33
    we can make these exponents positive
  • 00:27:35
    by flipping those variables individually
  • 00:27:38
    we don't need to change these two
  • 00:27:40
    because they already have a positive
  • 00:27:42
    exponent that is positive one
  • 00:27:45
    so we can say this is going to be 2
  • 00:27:47
    x to the positive 3
  • 00:27:49
    y to the positive 2 divided by
  • 00:27:52
    3
  • 00:27:53
    x to the positive 7 times y to the
  • 00:27:56
    positive 5
  • 00:27:58
    raised to the positive 4th power
  • 00:28:01
    so now it's easy to work with
  • 00:28:05
    so let's distribute the four
  • 00:28:11
    so this is going to be uh two to the
  • 00:28:14
    fourth power
  • 00:28:16
    three times four is twelve
  • 00:28:18
    two times four is eight
  • 00:28:22
    and this is going to be three to the
  • 00:28:23
    fourth power seven times four is twenty
  • 00:28:25
    eight
  • 00:28:26
    and five times four is twenty
  • 00:28:31
    now what is two to the fourth power
  • 00:28:34
    two to the fourth
  • 00:28:36
    two times two times two times two four
  • 00:28:38
    times that's sixteen
  • 00:28:41
    and three to the fourth we know it's 81.
  • 00:28:46
    now 12 minus 28
  • 00:28:49
    is negative 16
  • 00:28:52
    and 8 minus 20
  • 00:28:55
    is negative 12. so the final answer is
  • 00:28:57
    16
  • 00:28:58
    divided by
  • 00:29:00
    81 x raised to the 16th power
  • 00:29:03
    and y to the 12th power
  • 00:29:10
    now let's say if you have a question
  • 00:29:13
    like this
  • 00:29:14
    64
  • 00:29:16
    x to the nine
  • 00:29:18
    y to the
  • 00:29:19
    14
  • 00:29:21
    divided by 16
  • 00:29:24
    x to the fifth y to the eighth
  • 00:29:28
    raised to the third power
  • 00:29:31
    in this particular example you don't
  • 00:29:33
    want to do 64 raised to the third power
  • 00:29:35
    because that's a very very big number
  • 00:29:37
    and if you can't use calculators that's
  • 00:29:39
    not going to be a fun calculation
  • 00:29:42
    so when you're dealing with large
  • 00:29:43
    numbers it might be easier to divide
  • 00:29:45
    first before you raise it to the third
  • 00:29:47
    power
  • 00:29:49
    so let's not worry about the three for
  • 00:29:50
    now
  • 00:29:51
    64 divided by 16
  • 00:29:54
    is four
  • 00:29:56
    four is easier to deal with than 64.
  • 00:29:59
    nine minus five
  • 00:30:01
    is four
  • 00:30:04
    and fourteen minus eight
  • 00:30:06
    is six
  • 00:30:08
    and now we can raise this to the third
  • 00:30:10
    power
  • 00:30:13
    so it's going to be four cube
  • 00:30:15
    x to the twelfth and six times three is
  • 00:30:18
    eighteen
  • 00:30:19
    four times four times four is sixty-four
  • 00:30:21
    so the final answer is 64
  • 00:30:24
    x to the 12th y to the 18th
  • 00:30:30
    let's try this one
  • 00:30:32
    what is 2 times
  • 00:30:34
    what's 2x squared y cubed
  • 00:30:36
    raised to the third power
  • 00:30:38
    times
  • 00:30:40
    three x to the fourth
  • 00:30:43
    y to the negative fifth
  • 00:30:45
    raised to the second power
  • 00:30:49
    so let's focus on
  • 00:30:52
    the first one
  • 00:30:53
    it's going to be two to the third
  • 00:30:56
    and two times three is six three times
  • 00:30:58
    three is nine and for the second it's
  • 00:31:00
    going to be three squared
  • 00:31:02
    four times two is eight and negative
  • 00:31:04
    five times two is negative ten
  • 00:31:07
    now two to the third power two times two
  • 00:31:10
    times two is eight
  • 00:31:13
    and three squared
  • 00:31:15
    is nine
  • 00:31:20
    so now we can multiply
  • 00:31:22
    eight times nine and that's going to
  • 00:31:24
    give us 72
  • 00:31:28
    and then if we multiply x to the sixth
  • 00:31:30
    times x to the eight we know that six
  • 00:31:33
    plus eight
  • 00:31:34
    is equal to fourteen and finally a nine
  • 00:31:38
    y to the nine times y to negative ten
  • 00:31:40
    we know that nine plus negative ten
  • 00:31:43
    or nine minus ten is negative 1.
  • 00:31:46
    so therefore the final answer is 72
  • 00:31:49
    x to the 14th power
  • 00:31:52
    divided by
  • 00:31:53
    y to the first power or just simply y
  • 00:32:01
    try this one
  • 00:32:03
    three x
  • 00:32:05
    to the third y to the negative fourth
  • 00:32:08
    raised to the negative three
  • 00:32:11
    times
  • 00:32:12
    six x to the fifth power
  • 00:32:18
    y to the
  • 00:32:19
    negative two
  • 00:32:21
    raised to the second power feel free to
  • 00:32:23
    pause the video and work on this example
  • 00:32:26
    so this is three raised to the negative
  • 00:32:27
    3
  • 00:32:28
    and 3 times negative 3 is negative 9.
  • 00:32:32
    negative 4 times negative 3 is positive
  • 00:32:34
    12
  • 00:32:36
    and 5 times 2 is 10
  • 00:32:38
    negative 2 times 2 is negative 4.
  • 00:32:46
    now what exactly is three to the
  • 00:32:47
    negative third power
  • 00:32:49
    well first what's three to the third
  • 00:32:52
    three times three times three is
  • 00:32:53
    twenty-seven
  • 00:32:54
    therefore three to the minus three must
  • 00:32:56
    be one
  • 00:32:57
    divided by twenty seven
  • 00:33:01
    so what we have now is one over twenty
  • 00:33:04
    seven
  • 00:33:05
    times x to the negative nine
  • 00:33:07
    y to the twelve
  • 00:33:09
    6 squared
  • 00:33:10
    is 36
  • 00:33:14
    and now let's add the exponents
  • 00:33:17
    right now 1 over 27 times 36 is
  • 00:33:20
    basically
  • 00:33:21
    just 36 over 27.
  • 00:33:25
    negative nine plus ten
  • 00:33:27
    that's equal to positive one
  • 00:33:30
    and
  • 00:33:31
    twelve plus negative four
  • 00:33:33
    is positive eight
  • 00:33:35
    so this is the answer but actually we
  • 00:33:37
    can reduce it
  • 00:33:38
    because 36 and 27
  • 00:33:42
    are divisible by 9. 36 is 9 times 4
  • 00:33:48
    and 27 is 9 times 3.
  • 00:33:52
    so the final answer
  • 00:33:53
    is four x
  • 00:33:56
    y to the eighth divided by three
  • 00:34:03
    now it's time for a mix review for each
  • 00:34:05
    of these questions pause the video and
  • 00:34:07
    work on the examples
  • 00:34:09
    so what is 5 x squared
  • 00:34:12
    times 4 x cubed go ahead and try that
  • 00:34:15
    one
  • 00:34:16
    so 5 times 4 is 20
  • 00:34:19
    x squared times x cubed
  • 00:34:21
    two plus three is five
  • 00:34:23
    that's the answer for the first one
  • 00:34:27
    now what about this one what is three a
  • 00:34:30
    to the third power
  • 00:34:32
    b squared times five a to the negative
  • 00:34:36
    six
  • 00:34:38
    b to the negative fourth
  • 00:34:41
    so let's begin by multiplying three
  • 00:34:43
    times five three times five is fifteen
  • 00:34:46
    and then let's add 3 plus negative 6
  • 00:34:50
    which is negative 3
  • 00:34:53
    and then let's add 2 and negative 4
  • 00:34:56
    which is negative 2.
  • 00:34:58
    and our last step is to get rid of the
  • 00:34:59
    negative exponents
  • 00:35:01
    by moving a and b to the bottom so it's
  • 00:35:03
    15 divided by a to the third and b
  • 00:35:06
    squared
  • 00:35:10
    now what about this one
  • 00:35:12
    two y to the third
  • 00:35:14
    raised to the negative fourth what's the
  • 00:35:17
    answer
  • 00:35:19
    so let's distribute the negative four so
  • 00:35:21
    it's two to the negative four
  • 00:35:22
    and three times negative four is
  • 00:35:24
    negative twelve
  • 00:35:26
    so we can move everything to the bottom
  • 00:35:28
    so this is one divided by two to the
  • 00:35:30
    fourth y to the twelfth
  • 00:35:33
    two to the fourth is sixteen so the
  • 00:35:34
    final answer is one divided by sixteen
  • 00:35:37
    y to the twelve
  • 00:35:46
    now what about this one
  • 00:35:50
    we can move the three to the top so it's
  • 00:35:52
    going to be three to the positive four
  • 00:35:55
    and three to the fourth
  • 00:35:57
    is three times three times three times
  • 00:35:59
    three
  • 00:36:00
    which is nine times nine and that's
  • 00:36:03
    eighty one
  • 00:36:14
    go ahead and try this one
  • 00:36:17
    now what we could do is take the x move
  • 00:36:19
    it to the bottom
  • 00:36:20
    so it's 1 divided by 4
  • 00:36:22
    x to the fifth times x cubed and we can
  • 00:36:25
    add five plus three which is eight
  • 00:36:28
    so the final answer is one over four
  • 00:36:30
    x to the eighth power
  • 00:36:37
    what about this one y to the
  • 00:36:40
    negative seven
  • 00:36:41
    times y to the fourth
  • 00:36:43
    times y to the sixth
  • 00:36:47
    so let's multiply these two first
  • 00:36:50
    four plus six is equal to ten
  • 00:36:54
    and now we can multiply
  • 00:36:56
    these two now negative seven plus ten is
  • 00:36:59
    positive three
  • 00:37:00
    so this is the answer
  • 00:37:19
    now the first thing we should do in this
  • 00:37:20
    problem
  • 00:37:21
    is distribute the six
  • 00:37:23
    two times six is twelve
  • 00:37:26
    four times six is twenty four
  • 00:37:28
    and here the exponent we don't have one
  • 00:37:31
    so it's a one
  • 00:37:32
    which means this is still just two x
  • 00:37:34
    times y cubed
  • 00:37:38
    so this two will remain the same it's
  • 00:37:40
    like an invisible one here two times one
  • 00:37:42
    is simply two
  • 00:37:45
    x to the first power times x to the
  • 00:37:46
    twelfth one plus twelve
  • 00:37:49
    is 13
  • 00:37:52
    and finally 3 plus 24
  • 00:37:55
    is 27
  • 00:37:57
    so this is the answer
  • 00:38:05
    try this one
  • 00:38:07
    4 x cubed
  • 00:38:10
    times y
  • 00:38:11
    divided by 16
  • 00:38:14
    x to the fifth power
  • 00:38:16
    y cubed raised to the third power
  • 00:38:20
    so we can either divide first or we can
  • 00:38:22
    apply the exponent now do you know what
  • 00:38:24
    16 to the third power is what's 16 times
  • 00:38:27
    16 times 16
  • 00:38:28
    it's a very big number so we're going to
  • 00:38:30
    divide first now what is 4 divided by 16
  • 00:38:34
    if you're not sure divided backwards let
  • 00:38:36
    me give an example
  • 00:38:38
    let's say if you have
  • 00:38:40
    4 over 20
  • 00:38:41
    if you divide it backwards 20 divided by
  • 00:38:43
    4 is 5 but since you divide it backwards
  • 00:38:46
    you need to write it as 1 over 5.
  • 00:38:49
    if you think about it 4 over 20 is the
  • 00:38:51
    same as 4 over
  • 00:38:53
    4 times 5 because 4 times 5 is 20.
  • 00:38:57
    you can cancel a 4
  • 00:38:59
    and this leaves you with 1 over 5.
  • 00:39:01
    now the reason why you get a 1 on top is
  • 00:39:03
    because 4 divided by 4 is 1.
  • 00:39:07
    so 4 over 16 if we divide it backwards
  • 00:39:10
    16 divided by 4 is 4.
  • 00:39:12
    so 4 over 16 must be 1 over 4.
  • 00:39:17
    now let's subtract backwards
  • 00:39:19
    5 minus 3 is going to give us a 2 on the
  • 00:39:22
    bottom
  • 00:39:24
    if you subtract 4 it's 3 minus 5 is
  • 00:39:27
    negative two on top but if you move to
  • 00:39:28
    the bottom it's positive two
  • 00:39:31
    three minus one
  • 00:39:33
    is going to be two on the bottom
  • 00:39:36
    now keep in mind another way to see this
  • 00:39:38
    if you have x cubed divided by x to the
  • 00:39:39
    fifth
  • 00:39:40
    in your mind imagine that you have three
  • 00:39:42
    x's on top
  • 00:39:44
    and five x variables on the bottom
  • 00:39:47
    so you can cancel
  • 00:39:49
    three of them which leaves two on the
  • 00:39:50
    bottom
  • 00:39:52
    and that's what we have this number
  • 00:39:57
    so the same applies for
  • 00:40:00
    y to the first divided by y to the third
  • 00:40:05
    so we can cancel a y
  • 00:40:08
    leaving us two y variables on the bottom
  • 00:40:09
    which we can see here
  • 00:40:16
    so now we can raise everything to the
  • 00:40:17
    third power
  • 00:40:20
    four to the third
  • 00:40:22
    is 64.
  • 00:40:23
    and 2 times 3 is 6.
  • 00:40:26
    so this is the final answer
  • 00:40:29
    in this case it was easier to divide
  • 00:40:31
    first
  • 00:40:32
    than to use the exponent sometimes it's
  • 00:40:34
    easier to distribute the exponent than
  • 00:40:36
    to divide you have to decide which is
  • 00:40:39
    more convenient
  • 00:41:01
    go ahead take a minute and try this one
  • 00:41:05
    so do you think we should distribute the
  • 00:41:06
    negative 2 or simply to divide first
  • 00:41:11
    in this case we have small numbers so it
  • 00:41:12
    really doesn't matter
  • 00:41:14
    so this time
  • 00:41:16
    let's flip it first before we distribute
  • 00:41:18
    the negative two
  • 00:41:20
    so it's going to be three
  • 00:41:22
    x
  • 00:41:23
    y squared
  • 00:41:25
    z to the fifth power divided by negative
  • 00:41:28
    two x cubed
  • 00:41:30
    y to the third z to the zero power
  • 00:41:34
    and then now it's going to be squared
  • 00:41:38
    three squared is
  • 00:41:40
    nine
  • 00:41:41
    one times two is two
  • 00:41:43
    two times two is four
  • 00:41:45
    and five times two is ten
  • 00:41:47
    z to the 0 power is 1 so we could just
  • 00:41:50
    ignore this
  • 00:41:52
    now what's negative 2 squared
  • 00:41:54
    negative 2 squared is basically negative
  • 00:41:57
    2 times negative 2 which is positive 4.
  • 00:42:00
    so we're going to have a 4 on the bottom
  • 00:42:02
    and then 3 times 2 is 6
  • 00:42:05
    which means this is also 6 as well
  • 00:42:08
    so now we can subtract the exponents
  • 00:42:18
    we can't reduce nine over four
  • 00:42:21
    now for this one we can subtract it
  • 00:42:22
    backwards six minus two is going to give
  • 00:42:25
    us a four on the bottom
  • 00:42:27
    and six minus four
  • 00:42:30
    we'll put a two on the bottom
  • 00:42:32
    and the z to the ten will remain the
  • 00:42:33
    same
  • 00:42:34
    since there's nothing to subtract it
  • 00:42:36
    with
  • 00:42:37
    so this is the answer
  • 00:42:41
    here's a question for you
  • 00:42:43
    what is four to the third power
  • 00:42:47
    times four to the fifth power
  • 00:42:49
    what is the answer
  • 00:42:55
    think about it
  • 00:42:57
    now is it 16 to the eighth power
  • 00:43:01
    because 4 times 4 is 16 and 3 plus 5 is
  • 00:43:05
    8.
  • 00:43:06
    is this the answer correct
  • 00:43:08
    the answer is it's not correct
  • 00:43:12
    you can't multiply the bases
  • 00:43:14
    and add the exponents at the same time
  • 00:43:16
    you have to do one or the other
  • 00:43:19
    so in order to add the exponents the
  • 00:43:22
    bases must be the same
  • 00:43:24
    so if x cubed times x to the fifth
  • 00:43:27
    is equal to x to the eighth notice that
  • 00:43:28
    we didn't change x
  • 00:43:31
    four cubed times four to the fifth
  • 00:43:34
    is eight
  • 00:43:35
    to the eighth power
  • 00:43:39
    now here's a question for you
  • 00:43:41
    what is 4 times 4
  • 00:43:45
    now you know instinctively that 4 times
  • 00:43:48
    4 is 16
  • 00:43:49
    but now what are the exponents
  • 00:43:51
    4 is basically 4 to the first power 16
  • 00:43:54
    is 16 to the first
  • 00:43:56
    so notice that we kept the exponents the
  • 00:43:58
    same
  • 00:43:59
    if you keep the exponents the same
  • 00:44:01
    you are allowed to multiply the bases
  • 00:44:04
    so therefore
  • 00:44:06
    what is seven to the third times eight
  • 00:44:10
    to the third
  • 00:44:12
    so notice that the exponents are the
  • 00:44:14
    same which means we can multiply seven
  • 00:44:16
    times eight
  • 00:44:17
    and it's going to be 56 but the exponent
  • 00:44:20
    must remain the same so it's 56 cubed
  • 00:44:23
    so let me give you some examples on this
  • 00:44:26
    so what is five to the fourth times five
  • 00:44:28
    to the seventh
  • 00:44:31
    what is nine to the fifth times six to
  • 00:44:34
    the fifth
  • 00:44:36
    what is eight cubed times eight to the
  • 00:44:38
    eleventh
  • 00:44:40
    and what is seven
  • 00:44:42
    to the fourth
  • 00:44:44
    times
  • 00:44:48
    three to the fourth
  • 00:44:51
    so notice that the bases are the same
  • 00:44:53
    which means we can add the exponents
  • 00:44:55
    four plus seven is eleven
  • 00:44:58
    in the second example notice that the
  • 00:45:00
    exponents are the same which means we
  • 00:45:02
    can multiply the bases 9 times 6 is 54
  • 00:45:06
    but the exponent will remain 5.
  • 00:45:10
    now what about the next one
  • 00:45:12
    what is 8 cubed times 8 to the 11th
  • 00:45:15
    now the bases are the same so we can add
  • 00:45:18
    the exponents
  • 00:45:20
    3 plus 11 is 14.
  • 00:45:23
    and what about
  • 00:45:25
    the last one
  • 00:45:26
    the exponents are the same so we can
  • 00:45:28
    multiply the bases 7 times 3 is 21
  • 00:45:31
    and the exponent will stay 4.
  • 00:45:34
    so hopefully this helps you to
  • 00:45:35
    understand
  • 00:45:36
    the laws of exponents and how they work
  • 00:45:40
    so what if we get an example
  • 00:45:43
    where
  • 00:45:44
    the base any exponents are different
  • 00:45:48
    so notice that the bases are not the
  • 00:45:50
    same eight and four
  • 00:45:52
    and the exponents are not the same
  • 00:45:54
    either four and five
  • 00:45:56
    so what can we do
  • 00:45:58
    what we can't do we can't say 8 times 4
  • 00:46:01
    is 32 and add a 4 plus 5 and say it's 9.
  • 00:46:04
    that will not work do not do that
  • 00:46:07
    so in a situation like this
  • 00:46:09
    you need to make something equal you can
  • 00:46:11
    either change the base to a different
  • 00:46:13
    number or change the exponent until it's
  • 00:46:15
    the same
  • 00:46:16
    in this example
  • 00:46:18
    it's easier to change
  • 00:46:20
    the base into a common base
  • 00:46:24
    what number goes into 8 and 4.
  • 00:46:28
    eight and four are multiples of what
  • 00:46:30
    number
  • 00:46:33
    that number is two
  • 00:46:36
    so two is a common base of eight and
  • 00:46:38
    four
  • 00:46:41
    two times two
  • 00:46:42
    is four which means that four
  • 00:46:44
    is equal to two squared
  • 00:46:47
    two times two times two is eight
  • 00:46:49
    so it requires three twos to get to
  • 00:46:52
    eight
  • 00:46:52
    so which means that we can replace eight
  • 00:46:56
    with two to the third power
  • 00:46:58
    now we have to keep the four
  • 00:47:01
    and we can replace four
  • 00:47:03
    with two squared because four is equal
  • 00:47:05
    to two squared
  • 00:47:12
    so now whenever you raise one exponent
  • 00:47:14
    to another you need to multiply
  • 00:47:17
    three times four is twelve so this is
  • 00:47:20
    two to the twelfth power
  • 00:47:22
    and 2 times 5 is 10.
  • 00:47:27
    so now
  • 00:47:28
    notice that since we have a common base
  • 00:47:30
    we can add the exponents
  • 00:47:33
    12 plus 10
  • 00:47:34
    is 22 so this is 2 to the 22nd power
  • 00:47:38
    now if we want to
  • 00:47:40
    we can write this answer
  • 00:47:42
    using base 4 or base 8.
  • 00:47:46
    so let me show you how to do it
  • 00:47:54
    so let's say if we want to write the
  • 00:47:55
    answer
  • 00:47:56
    using base 4.
  • 00:47:59
    now we know that 4 is equal to 2 squared
  • 00:48:02
    so we got to take out 2 squared from it
  • 00:48:04
    22 is basically 2 times 11.
  • 00:48:09
    so we can write this as 2 squared
  • 00:48:12
    raised to the 11th power because 2 times
  • 00:48:14
    11 is 22 and 2 squared is 4. so this
  • 00:48:17
    answer is equivalent to 4 to the 11th
  • 00:48:19
    power
  • 00:48:21
    now what if we want to change it into
  • 00:48:23
    base 8
  • 00:48:24
    can we do that
  • 00:48:26
    we know 8 is 2 to the third power
  • 00:48:29
    and
  • 00:48:30
    3 times what number
  • 00:48:32
    is 22
  • 00:48:37
    so what number goes inside
  • 00:48:39
    to figure it out divide it it's going to
  • 00:48:41
    be this number divided by that number or
  • 00:48:44
    simply 22 over 3.
  • 00:48:47
    so we can say that
  • 00:48:49
    this is 2 to the third
  • 00:48:51
    times or raised to the 22 over the third
  • 00:48:54
    power because 3 times 22 over 3 is 22
  • 00:48:59
    and now we can replace 2 to the third
  • 00:49:01
    with eight
  • 00:49:02
    so this is eight
  • 00:49:04
    raised to the 22 over three
  • 00:49:17
    now let's try this example
  • 00:49:18
    9 to the fifth power
  • 00:49:21
    times 27 raised to the fourth power
  • 00:49:25
    now we know that 9 is 3 squared and 27
  • 00:49:29
    is 3 to the third
  • 00:49:30
    so let's replace 9 with 3 to the third
  • 00:49:34
    and let's replace 27
  • 00:49:36
    i mean i take that back
  • 00:49:38
    we need to replace 9 with 3 squared it's
  • 00:49:40
    very easy to make a mistake if you're
  • 00:49:41
    not too careful
  • 00:49:43
    let's replace 27 with three to the third
  • 00:49:48
    so now three squared
  • 00:49:51
    raised to the fifth power two times five
  • 00:49:53
    is ten
  • 00:49:54
    and three times four is twelve
  • 00:49:57
    so this is 3 to the 22nd power
  • 00:50:01
    which
  • 00:50:02
    if we want to convert it to base 9
  • 00:50:06
    that's 3 squared raised to the 11th
  • 00:50:08
    which is 9 to the 11th power
  • 00:50:11
    so i'm not going to change it to 27
  • 00:50:13
    because we're going to get a fractional
  • 00:50:14
    exponent so you can leave your answer
  • 00:50:16
    like this
  • 00:50:17
    or like this
  • 00:50:21
    now let's talk about radicals
  • 00:50:26
    how can you convert this expression into
  • 00:50:28
    a radical
  • 00:50:30
    x to the four thirds
  • 00:50:32
    is basically the cube root of x to the
  • 00:50:34
    fourth
  • 00:50:36
    x to the five over seven
  • 00:50:38
    is basically the seventh root of x to
  • 00:50:40
    the fifth power
  • 00:50:43
    x to the eight over five is basically
  • 00:50:44
    the fifth root
  • 00:50:46
    of x to the eighth so you need to be
  • 00:50:48
    able to convert
  • 00:50:49
    between
  • 00:50:51
    exponential form and radical form
  • 00:50:55
    so try these convert it back to
  • 00:50:57
    exponential form
  • 00:51:08
    so this is simply x to the four thirds
  • 00:51:10
    this is y to the nine fifths
  • 00:51:13
    and this is z to the eleven over four
  • 00:51:19
    now let's talk about how to simplify
  • 00:51:20
    radicals
  • 00:51:22
    what is the square root of x squared
  • 00:51:26
    and what is the square root of x to the
  • 00:51:28
    fourth
  • 00:51:29
    now if you don't see an index number
  • 00:51:32
    it's always assumed to be a two
  • 00:51:34
    so basically you're dividing
  • 00:51:36
    two by two which is one or simply x
  • 00:51:39
    now some textbooks whenever you have a
  • 00:51:42
    an even index number and an odd result
  • 00:51:45
    they may write it in absolute value just
  • 00:51:47
    so you know
  • 00:51:48
    you may see this so you may not but some
  • 00:51:50
    textbooks will leave it as the absolute
  • 00:51:51
    value of x
  • 00:51:53
    but in this video we're not going to be
  • 00:51:54
    too concerned with it
  • 00:51:57
    now in this case we're dividing 4 by 2.
  • 00:52:00
    so this is x squared
  • 00:52:02
    if you get an even exponent you don't
  • 00:52:03
    need to put it an absolute value because
  • 00:52:05
    it's always going to be positive
  • 00:52:09
    now what about the square root of x to
  • 00:52:11
    the sixth and the square root of x to
  • 00:52:12
    the eighth
  • 00:52:13
    what's the answer
  • 00:52:15
    the square root of x to the sixth is
  • 00:52:17
    gonna be six divided by two is just x to
  • 00:52:19
    the third
  • 00:52:20
    and the square root of x to the eighth
  • 00:52:22
    is x to the fourth
  • 00:52:26
    now what if we have the cube root of
  • 00:52:29
    let's say
  • 00:52:31
    x to the ninth and like the fourth root
  • 00:52:33
    of x to the 20
  • 00:52:35
    and the sixth root of x to the 42. how
  • 00:52:38
    can we simplify these expressions
  • 00:52:41
    so it may help to convert it back to its
  • 00:52:44
    exponential form
  • 00:52:47
    nine divided by three is four
  • 00:52:49
    and twenty divided by four is five
  • 00:52:53
    and for the last one we can write it as
  • 00:52:55
    42 divided by six which is seven
  • 00:53:04
    so what if we have the square root of an
  • 00:53:06
    odd number let's say like x to the ninth
  • 00:53:09
    what can we do in this case
  • 00:53:12
    well you can split it into two radicals
  • 00:53:15
    nine is basically eight plus one
  • 00:53:20
    the square root of x to the eight is
  • 00:53:22
    four because
  • 00:53:24
    eight divided by two is four
  • 00:53:26
    now the last one you really can't
  • 00:53:27
    simplify
  • 00:53:29
    you could write it as x to the one half
  • 00:53:30
    but i would simply leave it as the
  • 00:53:32
    square root of x
  • 00:53:33
    because
  • 00:53:35
    one is not divisible by two
  • 00:53:37
    you won't get an integer
  • 00:53:40
    so we'll leave it inside the radical so
  • 00:53:41
    this is the answer
  • 00:53:46
    let's try some more examples
  • 00:53:50
    try these
  • 00:53:53
    so for this one i'm going to write it as
  • 00:53:54
    the square root
  • 00:53:57
    of x to the 12th and the square root of
  • 00:53:59
    x
  • 00:54:00
    because 12 plus 1 is 13.
  • 00:54:02
    so this is going to be x to the 6 times
  • 00:54:04
    radical x
  • 00:54:06
    for the next one i'm going to write it
  • 00:54:07
    as the square root of x to the
  • 00:54:09
    times the square root of x to the first
  • 00:54:12
    six divided by two is three
  • 00:54:14
    and this is going to be the answer
  • 00:54:19
    now what if we have let's say the cube
  • 00:54:21
    root of x to the eleventh how can we
  • 00:54:24
    simplify
  • 00:54:26
    well the highest multiple of 3 that goes
  • 00:54:29
    into 11 is 9.
  • 00:54:34
    3 goes into 9 and 9 is under 11 so i'm
  • 00:54:37
    gonna write this as
  • 00:54:39
    x to the nine times x squared because
  • 00:54:42
    nine plus two is eleven
  • 00:54:44
    nine divided by three is three so the
  • 00:54:47
    final answer is x to the third times the
  • 00:54:49
    cube root of x squared
  • 00:54:55
    so what if we have let's say
  • 00:54:57
    the fourth root of x to the 17.
  • 00:55:02
    so what i would do is use 16 and 1
  • 00:55:05
    because 16 is the highest multiple
  • 00:55:08
    of 4 that's less than 17.
  • 00:55:13
    16 divided by 4
  • 00:55:15
    is 4.
  • 00:55:16
    so this is going to be x to the fourth
  • 00:55:18
    times the fourth root of x
  • 00:55:24
    try this one
  • 00:55:25
    the seventh root of x to the forty five
  • 00:55:29
    so i'm going to write it as the seventh
  • 00:55:30
    root of x to the forty two
  • 00:55:33
    and 45 minus 42 is three
  • 00:55:38
    42 divided by seven
  • 00:55:40
    is six
  • 00:55:41
    so it's x to the sixth times the seventh
  • 00:55:43
    root of x cubed
  • 00:55:46
    now sometimes you might have some
  • 00:55:48
    numbers inside as well for example let's
  • 00:55:50
    say if we have the square root
  • 00:55:53
    of eight
  • 00:55:54
    x to the sixth
  • 00:55:57
    now
  • 00:55:58
    what i would do is separate everything
  • 00:56:02
    a perfect square that goes into eight is
  • 00:56:04
    four so we have the square root of four
  • 00:56:07
    times root two
  • 00:56:08
    times root x to the sixth the square
  • 00:56:10
    root of four is two
  • 00:56:12
    the square root of x to the sixth is x
  • 00:56:14
    cubed
  • 00:56:15
    and we can't simplify radical two so
  • 00:56:17
    this is the answer
  • 00:56:21
    try this one
  • 00:56:22
    what is the square root of
  • 00:56:24
    75
  • 00:56:27
    x to the
  • 00:56:28
    ninth so a perfect square that goes into
  • 00:56:32
    75 is 25.
  • 00:56:33
    25 times 3 is 75 so i'm going to leave
  • 00:56:36
    it as square root 25 square root 3.
  • 00:56:39
    now 9 is an odd number so i'm going to
  • 00:56:41
    write it as x to the 8th and x
  • 00:56:44
    the square root of 25 is 5
  • 00:56:47
    and the square root of x to the 8
  • 00:56:49
    is x to the 4th
  • 00:56:52
    now these two we can't really simplify
  • 00:56:54
    but we can multiply them together so
  • 00:56:56
    it's going to be the square root of 3x
  • 00:56:59
    and that's the answer
  • 00:57:03
    now what about the cube root
  • 00:57:05
    of 16
  • 00:57:09
    x to the seventh
  • 00:57:13
    now the cube root of eight is two so we
  • 00:57:16
    wanna separate 16
  • 00:57:18
    into eight
  • 00:57:19
    and two
  • 00:57:21
    now six is a multiple of three so we're
  • 00:57:24
    gonna write it as x to the sixth
  • 00:57:26
    and x to the first power
  • 00:57:29
    the cube root of 8 is 2.
  • 00:57:31
    the cube root of x to the 6 is x squared
  • 00:57:34
    since 6 divided by 3 is 2.
  • 00:57:37
    now these two we can't simplify so we're
  • 00:57:38
    going to combine them so that's going to
  • 00:57:40
    be the cube root of 2x on the inside
  • 00:57:48
    try this one
  • 00:57:49
    the fourth root
  • 00:57:52
    of
  • 00:57:56
    48
  • 00:57:58
    x to the 11th
  • 00:58:01
    y to the
  • 00:58:06
    13. now
  • 00:58:09
    the fourth root of sixteen is two
  • 00:58:12
    and sixteen goes into forty eight
  • 00:58:14
    sixteen times three is forty eight so we
  • 00:58:16
    can say
  • 00:58:17
    this is the fourth root of sixteen
  • 00:58:20
    times the fourth root of three
  • 00:58:23
    and eleven let's separate into eight and
  • 00:58:25
    three
  • 00:58:30
    thirteen let's break it down into twelve
  • 00:58:33
    and one
  • 00:58:36
    the fourth root of 16 is two
  • 00:58:41
    the fourth root of x to the eight is x
  • 00:58:43
    squared eight divided by four is two
  • 00:58:46
    twelve divided by four is three
  • 00:58:50
    now everything else we can't simplify so
  • 00:58:52
    we're going to put it on the inside of
  • 00:58:53
    the radical
  • 00:58:56
    so on the inside is going to be 3
  • 00:58:59
    x cubed y
  • 00:59:00
    so this is our final answer
  • 00:59:04
    so that's it for this video and thanks
  • 00:59:06
    for watching it and hopefully you
  • 00:59:08
    understand exponents and how to use them
  • 00:59:10
    and have a great day
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  • exponents
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