Nth Roots and Rational Exponents (TEKS 2A.7G)

00:10:54
https://www.youtube.com/watch?v=8pRclfQ4_iQ

Resumo

TLDRThis video tutorial focuses on understanding nth roots and rational exponents as part of Algebra 2 curriculum. It starts by defining nth roots, explaining how to compute cube roots, and the importance of identifying positive and negative solutions when n is even. The tutorial moves on to rational exponents, where it describes how to convert them into radical form, clarifying the roles of the numerator and denominator in expressing the exponent and index respectively. Several examples guide the viewer through calculating roots and converting to radical form, enhancing comprehension through practical usage of a calculator. It highlights special considerations for negative numbers and provides a final recap of the conversions and evaluations.

Conclusões

  • 📚 Understand the definition of nth roots.
  • ✨ Cube root of 8 is 2, demonstrated with exponents.
  • 🔢 Positive and negative solutions appear for even roots.
  • 🧮 Use calculator functions for nth roots effectively.
  • 📖 Rational exponents can be rewritten in radical form.
  • ❌ No real solution for square roots of negative numbers.
  • 🔄 Negative exponents indicate inversion in calculations.
  • 🧩 Practical examples reinforce understanding of concepts.
  • 🌟 Always check your answers by reversing the operations.

Linha do tempo

  • 00:00:00 - 00:05:00

    The video introduces in routes and rational exponents, particularly focusing on the concept of roots in mathematics. It clarifies how to determine the value of n-th roots using examples like cube roots, highlighting the calculation of the cube root of -216 and the fourth root of 81. The speaker emphasizes that even roots yield both positive and negative solutions, exemplifying this with positive and negative 2 for the fourth root of 16, and discusses the non-existence of real solutions for negative square roots.

  • 00:05:00 - 00:10:54

    The discussion transitions to rational exponents, explaining the relationship between exponents and roots. The video details how to express square roots in fractional form (e.g., the square root of 'a' as a^(1/2)). Several examples follow, showcasing how to rewrite and evaluate expressions with rational exponents, such as 16^(3/2), demonstrating a systematic approach to simplifying these expressions before calculating their numerical values. The video concludes by encouraging engagement through likes and subscriptions.

Mapa mental

Vídeo de perguntas e respostas

  • What is an nth root?

    An nth root is the value that, when raised to the nth power, gives the original number.

  • How do you calculate the cube root of a number?

    Use your calculator to select the nth root function, or rewrite it as that number raised to the power of one-third.

  • What happens when the index n is even?

    When n is even, the root can have both a positive and negative solution.

  • What does a negative exponent indicate?

    A negative exponent indicates that the base should be taken as a fraction, moving it to the denominator.

  • How is a rational exponent written in radical form?

    A rational exponent like a^(m/n) can be rewritten as the nth root of a raised to the m power.

  • What is the square root of a negative number?

    The square root of a negative number does not have a real solution.

  • Can you have a rational exponent with a base of a variable?

    Yes, a variable can also have rational exponents, and the same rules apply.

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Rolagem automática:
  • 00:00:00
    good morning today we'll be working on
  • 00:00:02
    in route and rational exponents this is
  • 00:00:05
    a part 1 of two videos this is covering
  • 00:00:07
    teks 2a 7g from algebra 2 this is using
  • 00:00:12
    some Big Ideas algebra 2 math textbook
  • 00:00:15
    chapter 6.1 so we're going to first
  • 00:00:18
    start explaining wasn't what an in route
  • 00:00:20
    is you can extend the concept of a
  • 00:00:23
    square root to other types of roots so a
  • 00:00:25
    square root is and in through it means
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    that the root is 2 a cubed root would
  • 00:00:30
    mean that the N is 3 so for example
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    since 2 cubed equals 8
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    then the cubed root of 8 equals 2
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    one of the other ways we're going to
  • 00:00:41
    discuss today is this can be written 8
  • 00:00:43
    to the one-third equals 2 so using the
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    TI calculator one of the things we're
  • 00:00:48
    going to point out here is that this
  • 00:00:51
    button right here and depending on your
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    calculator will change the color we want
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    to select the enth root and so this
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    picture right here actually has a little
  • 00:01:01
    in the exponent so it looks like this
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    and so in order to select this we have
  • 00:01:09
    to click the control button first so
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    we're going to talk about taking the nth
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    root of different numbers so number one
  • 00:01:16
    says take the nth root of that number so
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    that means we're going to take the cube
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    root of negative 216 so we're going to
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    use our calculator and we're gonna do
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    control and select the in through button
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    basically what number to the third power
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    gives me negative 216 and the answer is
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    negative six we're gonna take the fourth
  • 00:01:42
    root of 81 okay now with any even in
  • 00:01:51
    numbers so if n is even and write this
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    down if n is even it has a positive and
  • 00:02:02
    negative solution now what do we mean it
  • 00:02:07
    has a positive or a negative solution
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    when my calculator if I go back to this
  • 00:02:11
    one and do the 4th root of 81 the
  • 00:02:13
    q leaders only going to give me the
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    positive solution it's gonna tell me
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    that the answer is a positive 3 but
  • 00:02:21
    let's think about that positive 3 to the
  • 00:02:24
    fourth power is 81 but go ahead and try
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    negative 3 in parenthesis to the 4th
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    power doesn't it also give you 81 it
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    does so whenever n is even you're going
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    to do the positive and the negative
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    solution by it again okay so number
  • 00:02:48
    number 3 we're going to take the 4th
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    root of 16 what number to the 4th power
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    gives you 16 and don't forget if n is
  • 00:03:00
    even it's gonna have a positive and
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    negative solution so the 4th root of 16
  • 00:03:07
    is a positive 2 but it's also a negative
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    2 okay this 2 here that's the square
  • 00:03:16
    root guys I don't actually have to put
  • 00:03:18
    the 2 if the index so that's that n is
  • 00:03:22
    called an index if the index is a 2 you
  • 00:03:31
    don't have to write it that's what a
  • 00:03:32
    square root is what is the square root
  • 00:03:34
    of negative 49 tried in your calculator
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    it actually comes back as there's not a
  • 00:03:39
    real solution there is no number that
  • 00:03:42
    you can square that'll give you a
  • 00:03:44
    negative number so something like this
  • 00:03:46
    would come back no solution the next one
  • 00:03:53
    I'm going to take the cube root of
  • 00:03:56
    negative 125 so what number to the third
  • 00:04:00
    power gives me negative 125 the answer
  • 00:04:03
    is negative 5 and guys the way you check
  • 00:04:09
    that is negative 5 in parentheses to the
  • 00:04:12
    third power it should give you negative
  • 00:04:15
    125 that is correct
  • 00:04:19
    number 6 we're going to take the fifth
  • 00:04:21
    root of 243 what is the 5th
  • 00:04:27
    root of 243 well the answer is three and
  • 00:04:32
    so we check it does three to the fifth
  • 00:04:35
    power
  • 00:04:36
    does it give us 243 and it does so
  • 00:04:41
    that's how we take the nth root of
  • 00:04:43
    problems now let's get into rational
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    exponents these ones are gonna be really
  • 00:04:48
    really important I want to explain where
  • 00:04:52
    these letters and numbers come from so
  • 00:04:55
    when you take the square root so we're
  • 00:04:57
    taking the square root of a that number
  • 00:05:01
    on the outside is an imaginary two we
  • 00:05:03
    don't write a two but we see it what it
  • 00:05:06
    is there and it has an exponent of a 1
  • 00:05:08
    so the square root of a could be really
  • 00:05:11
    written as a to the one half so the way
  • 00:05:15
    the fractions work that fraction on the
  • 00:05:18
    top is gonna be your exponent and the
  • 00:05:21
    bottom is going to be your index I wrote
  • 00:05:25
    that better for you so on the numerator
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    is the exponent that is gonna stay
  • 00:05:30
    outside of your parenthesis the index is
  • 00:05:33
    what's gonna go inside your radical it's
  • 00:05:36
    called the index so this number right
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    here if it's a square root that's a 2 we
  • 00:05:42
    don't see the two but if it's a cube
  • 00:05:43
    root we'll see a 3 there if it's the 4th
  • 00:05:46
    root we'll see a 4 there now negatives
  • 00:05:49
    we're not gonna see a ton of them they
  • 00:05:51
    work almost exactly the same the only
  • 00:05:54
    difference is if you have a negative so
  • 00:05:56
    in algebra 1 we talked about it's an
  • 00:06:00
    exponents a negative it's not happy
  • 00:06:02
    there or like it doesn't belong there so
  • 00:06:04
    if it's a negative you're actually gonna
  • 00:06:06
    put it to the bottom and then it belongs
  • 00:06:08
    there and it turns positive ok so if
  • 00:06:11
    it's a negative we move it to the bottom
  • 00:06:12
    and do exactly what we see so before I
  • 00:06:15
    even move on to the next page so it's
  • 00:06:17
    easier to see while we're on this if I
  • 00:06:20
    have a letter B and I say it's to the
  • 00:06:24
    2/5 power how can we rewrite that well
  • 00:06:30
    we're gonna take the 5th root of B and
  • 00:06:35
    all of that's going to be squared
  • 00:06:38
    okay so the five is the index and that
  • 00:06:43
    index is going to go here the top is my
  • 00:06:50
    exponent and it's gonna stay an exponent
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    okay we are going to rewrite in radical
  • 00:06:58
    form and then we're going to evaluate so
  • 00:07:01
    16 to the three-halves is gonna be
  • 00:07:03
    rewritten 16 is my base I'm gonna put a
  • 00:07:08
    radical what goes on the inside of the
  • 00:07:11
    index and what goes on the outside as
  • 00:07:14
    the exponent well as we wrote the index
  • 00:07:19
    is the bottom number the denominator if
  • 00:07:21
    it's a two I do not have to write a two
  • 00:07:24
    in my index that's automatically a
  • 00:07:26
    square root and this is gonna be raised
  • 00:07:28
    to the third power what is the square
  • 00:07:31
    root of 16 which we knew positive
  • 00:07:33
    solutions so the square root of 16
  • 00:07:36
    becomes 4 4 cubed 4 times 4 times 4 is
  • 00:07:42
    gonna be 64 so when you're showing the
  • 00:07:47
    work you're gonna rewrite it first and
  • 00:07:49
    try not using a calculator so number two
  • 00:07:53
    we're gonna start by rewriting it now it
  • 00:07:55
    has a negative exponent if it's a
  • 00:07:57
    negative exponent we're gonna rewrite it
  • 00:07:59
    as 1 over 32 to the 3/5 power okay so on
  • 00:08:07
    my denominator my 32 is going to go
  • 00:08:11
    inside of my radical I'm gonna have a 5
  • 00:08:14
    on the outside that is my index and this
  • 00:08:17
    is going to be raised to the third power
  • 00:08:19
    what is the fifth root of 32 must be a
  • 00:08:23
    really small number what number to the
  • 00:08:24
    fifth power gives us 32 it's 2 so 1 over
  • 00:08:29
    2 cubed 2 times 2 times 2 the answer is
  • 00:08:34
    1/8 again this time we have a 4 4 is my
  • 00:08:41
    base the denominators 2 which makes it a
  • 00:08:44
    square root to the fifth power and guys
  • 00:08:48
    what you're not seeing what you never
  • 00:08:50
    have to do is you don't to put the 2
  • 00:08:52
    they're a square-root already has it too
  • 00:08:55
    well what's the square root of 4 the
  • 00:08:58
    answer is 2 and what is 2 to the fit
  • 00:09:01
    power the answer is 32 okay the next one
  • 00:09:09
    it's a negative exponent if it's a
  • 00:09:12
    negative exponent I'm going to rewrite
  • 00:09:13
    it first with a positive exponent if
  • 00:09:16
    it's negative you're gonna move it to
  • 00:09:18
    the denominator 9 to the 1/2 power means
  • 00:09:22
    it's actually the square root of 9 what
  • 00:09:26
    is the square root of 9 the answer is 3
  • 00:09:29
    now why was it the square root of 9
  • 00:09:31
    because every single problem can have a
  • 00:09:34
    1 exponent every single problem can have
  • 00:09:36
    a 2 index and you don't have to show the
  • 00:09:38
    1 or the 2 okay so anything to the 1/2
  • 00:09:42
    power is just a square root number 581
  • 00:09:48
    and we're gonna do the 4th root of 81
  • 00:09:52
    and whatever we get to the 3rd power
  • 00:09:56
    the 4th root of 81 is 3 so 3 to the 3rd
  • 00:10:02
    power 3 times 3 times 3 is 27 the last
  • 00:10:09
    one 1 is my base it goes inside of my
  • 00:10:13
    radical I'm taking the 8th root of 1 and
  • 00:10:19
    I'm gonna raise it to the seventh power
  • 00:10:21
    okay what's the 8th root of 1 what
  • 00:10:24
    number to the 8th power gives you 1 1 1
  • 00:10:28
    to the 7th is 1
  • 00:10:35
    hopefully this video helped you
  • 00:10:37
    understand a little bit of how to
  • 00:10:39
    rewrite rational exponents as radicals
  • 00:10:43
    and if you like this video if it helped
  • 00:10:45
    you at all please give it a thumbs up
  • 00:10:47
    you can subscribe to this channel well I
  • 00:10:49
    will be posting more educational videos
  • 00:10:51
    thanks and have a great day
Etiquetas
  • nth roots
  • rational exponents
  • Algebra 2
  • calculations
  • radical form
  • negative exponent
  • square root
  • cube root
  • calculator use
  • mathematical concepts