Rational Exponents

00:05:40
https://www.youtube.com/watch?v=tXuyPiYvGZ4

Résumé

TLDRThe video explains rational exponents, which are exponents that can be expressed as fractions. It details how to interpret a fractional exponent, where the numerator represents the power and the denominator represents the root. For example, 4^(1/2) is equal to the square root of 4, simplifying to 2. The tutorial includes examples like 16^(1/2) and 32^(3/2) to reinforce this concept. It also addresses negative exponents and the equivalence between radical and exponential forms, highlighting that expressions like the 4th root of 5^7 can be written as 5^(7/4). Overall, viewers learn the importance and application of understanding rational exponents.

A retenir

  • 📘 Rational exponents are fractions representing powers and roots.
  • 🔍 The numerator indicates the power, and the denominator indicates the root.
  • 🟠 4^(1/2) = √4 = 2.
  • 🔸 16^(1/2) = √16 = 4.
  • 🔄 Negative exponents imply taking the reciprocal.
  • ⭐ 32^(3/2) = √(32^3).
  • 🔁 Recognize equivalence between radical and exponential forms.
  • ➕ Example: 4th root of 5^7 = 5^(7/4).
  • 🤔 Knowing how to convert between forms simplifies calculations.

Chronologie

  • 00:00:00 - 00:05:40

    This lesson focuses on rational exponents, specifically examining fractional exponents such as 4 raised to the power of 1/2. The exponent's numerator represents the power, while the denominator indicates the root, leading to the conclusion that 4^(1/2) is equivalent to the square root of 4, which simplifies to 2. Continuing with examples, 16^(1/2) also simplifies to 4, illustrating the general rule that b^(m/n) equals the n-th root of b raised to the m-th power. Further illustrating this, 32^(3/2) is broken down into the square root of 32 raised to the power of 3, demonstrating how fractional exponents operate similarly to integer exponents. The lesson concludes with the handling of negative exponents, where 18^(-10/7) is transformed into 1/(18^(10/7)), reinforcing the concept that both radical forms and fractional exponent forms can represent the same value.

Carte mentale

Vidéo Q&R

  • What are rational exponents?

    Rational exponents are exponents that are fractions, where the numerator indicates the power and the denominator indicates the root.

  • How do you interpret a fractional exponent?

    A fractional exponent like a/b means you take the b-th root of the number raised to the power of a.

  • What does 4^(1/2) equal?

    4^(1/2) equals the square root of 4, which simplifies to 2.

  • What is 32^(3/2)?

    32^(3/2) can be interpreted as the square root of 32 raised to the 3rd power.

  • How do you handle negative exponents?

    A negative exponent indicates the reciprocal; for example, 18^(-10/7) equals 1/(18^(10/7)).

  • What is the relation between radical and exponential form?

    The n-th root of a number can be expressed as the number to the power of 1/n.

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  • 00:00:00
    today we're going to be learning about
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    rational exponents so far the exponents
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    that we've been considering were
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    integers negative numbers zeroes
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    positive numbers but we never looked at
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    fractions as exponents for example if we
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    have something like 4 to the exponent 1
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    over 2 what would that be
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    it's certainly difficult to understand
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    what this would mean but there is
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    another equivalent form that might help
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    us to understand what this means
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    now this over here is equal to this so
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    notice how this exponent over here has a
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    numerator of 1 and a denominator of 2
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    what we end up with is square root of 4
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    2 the exponent 1 so once again you can
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    see that from the exponent the
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    denominator is what determined the
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    degree of the root of course since the
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    denominator is 2 we have a root of
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    degree 2 more commonly known as a square
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    root and since the numerator is 1 we are
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    looking for the square root of 4 2 the
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    exponent 1 and of course the square root
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    of 4 can be further simplified to 2
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    great so if we had something like 16 to
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    the exponent 1 over 2 then this would be
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    the square root of 16 the exponent 1 of
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    course that simplifies down to 4 and we
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    can say this in a general sense notice
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    here what is happening to be and what is
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    happening to a now with that in mind
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    let's try another example where we
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    specifically try to use this general
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    formula and you'll notice that it's
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    actually really easy to do so if we had
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    something like 32 to the exponent 3 over
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    2 that would just be the square root of
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    32 to the exponent 3 if it helps you to
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    understand exponents better another way
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    to do this would have been to see this 3
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    over 2 as the same thing as 3 times 1
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    over 2 after all 3 is equal to 3 over 1
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    and when you multiply the numerators and
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    the denominators you would end up at 3
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    over 2 now if you decide to see the
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    exponent as 3 times 1 over 2 then you
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    might also be able to see that this is
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    essentially a power of a power situation
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    where you have 32 to the exponent 3
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    bracket it to the exponent 1 over 2 and
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    now of course you see the similarities
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    between this example and our very first
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    example where we did 4 to the exponent 1
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    over 2 remember that when we raised 4 to
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    the exponent of 1 over 2 it was the same
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    thing as simply taking the square root
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    of 4 similarly if we're raising 32 to
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    the exponent 3 to the exponent of 1 over
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    2 then that would be the same as taking
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    the square root of this entire power
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    right here awesome so let's just do one
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    last question let's make this a little
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    bit more difficult but don't let that
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    fool you this isn't actually as
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    difficult as it might seem so we have
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    something like this we see some
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    fractions in the exponents well first
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    all we have to do is deal with the
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    operators negative 12 over 7 plus 2 over
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    7 well that's pretty simple that's just
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    going to be negative 10 over 7 now the
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    question is what do we do from here well
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    we have already learned what to do with
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    negative exponents 18 to the exponent
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    negative 10 over 7 is going to be the
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    exact same thing as 1 over 18 to the
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    exponent 10 over Sonne remember we can
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    just do one over and
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    that entire power dropping that negative
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    sign great now we're almost finished but
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    we did learn something today we can put
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    that 18 to the exponent 10 over 7 in a
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    different form
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    do we need to maybe not but if our
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    teacher wants us to then maybe we should
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    so in this situation we have 1 over 7th
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    root of 18 to the exponent 10 now
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    reversely if we saw something in its
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    radical form we should be able to find
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    out its equivalent exponential form as
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    well so if we had something like the 4th
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    root of 5 to the exponent 7 then of
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    course this can be translated to 5 to
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    the exponent 7 over 4 so it becomes
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    pretty obvious then that this formula
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    over here can help us out quite a bit
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    just remember that they are two
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    different ways to write the same thing
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    the same value so there you have it now
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    you know how to interpret powers that
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    have fractions as their exponents and we
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    will see you in our next video
Tags
  • Rational Exponents
  • Fractional Exponents
  • Square Root
  • Negative Exponents
  • Radical Form
  • Exponential Form
  • Power of a Power
  • Root
  • Numerator
  • Denominator