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Evaluating fractional exponents: fractional base

Sal shows how to evaluate (25/9)^(1/2) and (81/256)^(-1/4). Created by Sal Khan.

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  • blobby green style avatar for user 2020NinaF276
    At Sal does the reciprocal to make the fraction positive, is this the same as moving the entire fraction to the denominator and going from there? Thanks!
    (11 votes)
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  • mr pants teal style avatar for user Lucian
    If I a number does not pop out at me when I am trying to simplify a radical, is there a quicker way to figure out its answer besides guessing?
    (6 votes)
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  • orange juice squid orange style avatar for user Abraham George
    At , wouldn't the 5/3 raised to the 2 power be written like this (5/3)^2 rather than like 5^2/3? Shouldn't the parentheses be there?
    (7 votes)
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  • aqualine ultimate style avatar for user ankit.roy.us
    Couldn't you also find the prime factorization to find 256^1/4 or 81^1/4?
    (4 votes)
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  • blobby green style avatar for user David Yuxin
    why are you only looking for the principle root?
    (3 votes)
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  • aqualine seed style avatar for user Victoria Pallister
    What if you had a decimal number to the power of a negative exponent? for example 0.3 ^ -4
    (2 votes)
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    • duskpin sapling style avatar for user Vu
      I don't know if you are familiar with the exponent rules. We can use the negative and power exponent rules:
      Negative exponent: a^-1 = 1/ (a^1)
      Power rule: (a^m)^n = a^(mn)

      we can rewrite 0.3^-4 as 0.3^ (-1 * 4)
      Using power rule we get (0.3^4)^-1
      Using negative exponent rule we get 1/(0.3^4)
      Which equals to 1/ (0.3 * 0.3 * 0.3 * 0.3) = 1/0.0081
      (2 votes)
  • male robot hal style avatar for user Connor Chun
    How would I simplify a problem that is to the fourth root, but has a negative base? Something like (-625)^(3/4).
    (2 votes)
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    • leaf orange style avatar for user A/V
      First things first, depends on the expression. You would either get a solution with i, or you'd get a rational number. Let's take your example:
      (-625)^(3/4)

      Best way to approach this is to split it up, remember that:
      ab² = a²b² (² being an example)

      -1^(3/4) * 625^(3/4)
      Looking at 625, it can be broken down into:
      5⁴

      Using (aᵇ)ᶜ = aᵇᶜ:
      ((5)⁴)^(3/4) = 5³ = 125

      Your answer would be:
      125 * -1^(3/4)
      Which cannot be simplified further !

      Note that other cases might cannot simplify like this, it's simply a matter of breaking down the question.
      (2 votes)
  • blobby green style avatar for user April W Saclayan
    if it the problem is (125/27) ^ -2/3 how would you solve it?
    (2 votes)
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  • leafers ultimate style avatar for user Mark
    At Sal gets rid of the (-) sign by taking the reciprocal of the fraction, what I did was just divide the entire thing by 1, and then at the end I had 1/(3/4) which works out to 4/3 which is the right answer, but is there any particular reason why he chose to take the reciprocal instead?
    (1 vote)
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    • starky ultimate style avatar for user Kevin Monisit
      You're lucky that you got 4/3. But what you did disregards what is actually happening mathematically. Dividing by 1 won't work on everything.

      The fraction is 3/4 when it's reduced right? So the original statement was (3/4) ^ (-1/4) power. As you know, any number to any fractional power makes it the root of it.

      Like a ^ (1/2) is the sqrt(a), right? Thus, a ^ (1/3) would be the CUBE root of a.

      When we have a negative fraction as the power, we take the reciprocal because its the reversal of it. So a ^ (-1/2), would be 1 / sqrt(a). That means a/b ^ (-1/2) would be sqrt(b/a).

      The only reason why (256/81)^(1/4) equaled 4/3 was because it was taking the fourth root of 256 and 81, which so happens to equal 4/3, or 1 DIVIDED by (81/256), which is what you did.

      If he used a different fractional power, your answer would be wrong.
      (4 votes)
  • blobby green style avatar for user nickvanraden
    how do you answer the question when the 4th root of a number doesn't work out so squeaky clean. e.g., 81 1/4th power through prime factorization works out clean with 3, 4's. Seeing as how I don't know how to find advanced numbers 3rd 4th 5th roots etc. other than by a prime factorization how do I represent it if i end up with something that doesn't have the exact number of prime factors as the root that I am taking it too?
    (2 votes)
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Video transcript

Fractional exponents can be a little daunting at first, so it never hurts to do as many examples as possible. So let's do a few. What if we had 25/9, and we wanted to raise it to the 1/2 power? So we're essentially just saying, well, what is the principal square root of 25/9? So what number times itself is going to be 25/9? Well, we know 5 times 5 is 25, and 3 times 3 is 9. So why don't we just go with 5/3? Because notice, if you have 5/3 times 5/3, that is going to be 25/9. Or another way of saying this, that 5/3 squared is equal to 25/9. So 25/9 to the 1/2 is going to be equal to 5/3. Now let's escalate things a little bit. Let's take a really hairy one. Let's raise 81/256 to the negative 1/4 power. I encourage you to pause this and try this on your own. So what's going on here? This negative-- the first thing I always like to do is I want to get rid of this negative in the exponent. So let me just take the reciprocal of this and raise it to the positive. So I could just say that this is equal to 256/81 to the 1/4 power. And so now I can say, well, what number times itself times itself times itself is going to be equal to 256, and what number times itself times itself times itself-- did I say that four times? Well, what number, if I take four of them and multiply, do I get 81? And one way to think about it, this is going to be the same thing-- and we'll talk about this in more depth later on when we talk about exponent properties. But this is going to be the exact same thing as 256 to the 1/4 over 81 to the 1/4. You, in fact, saw it over here. This over here was the same thing as the square root of 25 over the square root of 9. Or 25 to the 1/2 over 9 to the 1/2. So we're just doing that over here. So one, we still have to think about what number this is. And this is a little bit of, there's no easy way to do this. You kind of have to just play around a little bit to come up with it. But 4 might jump out at you if you recognize that 16 times 16 is 256. We know that 4 to the fourth power, or you're about to know this, is 4 times 4 times 4 times 4. And 4 times 4 is 16, times 4 is 64, times 4 is equal to 256. So 4 to the fourth is 256, or we could say 4 is equal to 256 to the 1/4 power. Fair enough? Now what about 81? Well, 3 might jump out at you. We know that 3 to the fourth power is equal to 3 times 3 times 3 times 3, which is equal to 81. So 3 is equal to 81 to the 1/4. So this top number, 256 to the 1/4, that's just 4. 81 to the 1/4, that is just 3. So this right over here is going to be equal to 4/3.