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Evaluating fractional exponents

Sal shows how to evaluate 64^(2/3) and (8/27)^(-2/3). Created by Sal Khan.

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  • aqualine ultimate style avatar for user Marcia Vela
    At , can you explain why the reciprocal eliminates the negative sign? What does flipping the fraction do that makes it a positive?
    (32 votes)
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  • piceratops ultimate style avatar for user Stephen
    So is x^2/3 the same thing as finding the cube root of x, and then squaring it? (or vice versa)
    (16 votes)
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  • blobby green style avatar for user Keng.Herr
    I'm stuck on a problem.
    How would you simplify the following: (x^3)^(2/3)

    My first thought would be to multiply the exponents: 3/1 * 2/3 which would leave me with an exponent of 2. Can anyone confirm this answer for me?
    (6 votes)
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  • hopper jumping style avatar for user Kwaku Afriyie
    at did Sal mean numerator
    (5 votes)
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  • blobby green style avatar for user Kira Olver
    how do you make this solution work when you are working with negatives? Example: (-7776)^(2/5) or 〖4096〗^(-5/6)
    (3 votes)
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    • mr pink green style avatar for user David Severin
      With a negative number inside the root, you cannot take the root if it is even (the denominator of the fraction), but it if it is odd, then the answer will end up negative. 7776 = 6^5 (rather than going through factoring, I did 7776^(1/5) in calculator), so squaring we end up with (-6)^2 which ends up as 36. With a negative exponent, this causes the expression to reciprocate and change exponent to positive, so start with 1/(4096)^(5/6) = 1/4^5 = 1/1024.
      (6 votes)
  • leafers tree style avatar for user htom
    well what if something was like 1/2 to the power of 7 how would you
    solve that?
    (3 votes)
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  • blobby green style avatar for user Johnine Marie Casia
    I have a question..
    Why is (-9)^1/2 not a real number?
    (2 votes)
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  • winston default style avatar for user Aaron F
    What do do when a fraction is squared by a fraction?
    (3 votes)
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    • duskpin ultimate style avatar for user Achintya
      If a fraction is raised to the power of a fraction, for eg.
      if 2/7 is raised to the power of 7/9, then 2/7 would be under the root of 9, and 2/7 would be raised to the power of 7 when it's under.
      HOPE THIS HELPS,
      THANK YOU
      (2 votes)
  • starky ultimate style avatar for user Bob the Reaper
    How does one solve a number to a fraction power, such as six to the power of one eleventh? I still do not get it
    (1 vote)
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    • mr pink green style avatar for user David Severin
      Fractional powers, also called rational exponents, are a different way of writing roots of numbers, the numerator is the power of the term inside the root and the denominator is the power of the root. SO 6^(1/11) would be the same as the eleventh root of 6, written with a six inside the root sign and a small 11 on the crook of the root sign (√) which is sort of inside the V part of the root sign. There is nothing to solve unless you want an approximation which you can get by entering 6^(1/11) into a calculator.
      (5 votes)
  • female robot grace style avatar for user Anna
    If 2^1 is 2 and 2^2 is 4 is 2^3/2 3? and what about other powers of two? I mean I know not all integers can be expressed as a power of two (at least a rational one that we first deal with) but is there a power of two with some fraction as the exponent that equals 5,6, or 7?
    (4 votes)
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Video transcript

We've already seen how to think about something like 64 to the 1/3 power. We saw that this is the exact same thing as taking the cube root of 64. And because we know that 4 times 4 times 4, or 4 to the third power, is equal to 64, if we're looking for the cube root of 64, we're looking for a number that that number times that number times that same number is going to be equal to 64. Well, we know that number is 4, so this thing right over here is going to be 4. Now we're going to think of slightly more complex fractional exponents. The one we see here has a 1 in the numerator. Now we're going to see something different. So what I want to do is think about what 64 to the 2/3 power is. And here I'm going to use a property of exponents that we'll study more later on. But this property of exponents is the idea that-- let's say with a simple number-- if I raise something to the third power and then I were to raise that to, say, the fourth power, this is going to be the same thing as raising it to the 2 to the 3 times 4 power, or 2 to the 12th power, which you could also write as raising it to the fourth power and then the third power. All this is saying is, if I raise something to a power and then raise that whole thing to a power, it's the same thing as multiplying the two exponents. This is the same thing as 2 to the 12th. So we could use that property here to say, well, 2/3 is the same thing as 1/3 times 2. So we could go in the other direction. We could say, hey look, well this is going to be the same thing as 64 to the 1/3 power and then that thing squared. Notice, I'm raising something to a power and then raising that to a power. If I were to multiply these two things, I would get 64 to the 2/3 power. Now, why did I do this? Well, we already know what 64 to the 1/3 power is. We just calculated it. That's equal to 4. So we could say that-- and I'll write it in that same yellow color-- this is equal to 4 squared, which is equal to 16. So 64 to the 2/3 power is equal to 16. The way I think of it, let me find the cube root of 64, which is 4. And then let me square it. And that is going to get me to 16. Now I'll give you in even hairier problem. And I encourage you to try this one on your own before I work through it. So we're going to work with 8/27. And we're going to raise this thing to the-- and I'll try to color code it-- negative 2 over 3 power, to the negative 2/3 power. I encourage you to pause and try this on your own. Well the first thing I do whenever I see a negative exponent is to say, well, how can I get rid of that negative exponent? And I just remind myself, well, the negative exponent really just says, take the reciprocal of this to the positive exponent. I'm using a different color. I'm going to use that light mauve color. So this is going to be equal to 27/8. I just took the reciprocal of this right over here. It's equal to 27/8 to the positive 2/3 power. So notice, all I did, I got rid of the exponent and took the reciprocal of the base right over here. 8/27 is the base. Negative 2/3 is the exponent. Now, how can we handle this? Well, we've already seen that if I have a numerator to some power over a denominator to some power-- and this is another very powerful exponent property-- this is going to be the exact same thing as raising 27 to the 2/3 power-- to the 2 over 3 power-- over 8 to the 2/3 power. This is another very powerful exponent property. Notice, if I have something divided by something and I'm raising the whole thing to a power, I can essentially raise the numerator to that power over the denominator raised to that power. Now, let's think about what this is. Well just like we saw before, this is going to be the same thing. This is going to be the same thing as 27 to the 1/3 power and then that squared because 1/3 times 2 is 2/3. So I'm going to raise 27 to the 1/3 power and then square whatever that is. All this color coding is making this have to switch a lot of colors. This is going to be over 8 to the 1/3 power. And then that's going to be raised to the second power. Same thing we were doing in the denominator-- we raise 8 to the 1/3 and then square that. So what's this going to be? Well, 27 to the 1/3 power is the cube root of 27. It's some number-- that number times that same number times that same number is going to be equal to 27. Well, it might jump out at you already that 3 to the third is equal to 27 or that 27 to the 1/3 is equal to 3. So the numerator, we're going to end up with 3 squared. And then in the denominator, we are going to end up with-- well, what's 8 to the 1/3 power? Well, 2 times 2 times 2 is 8. So this is 8 to the 1/3 third is 2. Let me do that same orange color. 8 to the 1/3 is 2, and then we're going to square that. So this is going to simplify to 3 squared over 2 squared, which is just going to be equal to 9/4. So if you just break it down step by step, it actually is not too daunting.