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Evaluating mixed radicals and exponents

A worked example of calculating an expression that has both a radical and an exponent. In this example, we evaluate 6^(1/2)⋅(⁵√6)³. Created by Sal Khan.

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  • old spice man green style avatar for user Tasha
    Shouldn't the answer be 36^11/10 because 6*6=36 and not 6? If not then how did he get 6?
    (17 votes)
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  • leaf green style avatar for user Michael Martins
    I am in intermediate algebra/trig in COLLEGE and this is what I am learning; how is this Algebra I ? :(
    (10 votes)
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  • mr pants teal style avatar for user Jessica Chavarria
    How would you do this with different bases instead?
    (19 votes)
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  • cacteye yellow style avatar for user Br Paul
    , when we multiply 6^1/2 and 6^3/5 together, wouldn't that equal 36^11/10?
    (6 votes)
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  • leaf grey style avatar for user Mike Snow Carini
    i don't understand why the two sixes at the end don't get multiplied

    i was expecting the result to be 36 to the 11/10 power
    i ve seen this happen quite often and cant really come up with an answer, i ve seen someone asking this already, but i would like to have an intuitive definition for this process

    thanks so much for the help
    mike
    (6 votes)
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    • stelly blue style avatar for user Kim Seidel
      Exponents represent repetitive multiplication of a common base value (the 6). It may be easier to understand this by using simpler exponents.
      Consider 6^2 * 6^3
      6^2 = 6*6
      6^3 = 6*6*6
      Thus, 6^2 * 6^3 = 6*6 * 6*6*6 = 6^5
      The 6 is the value that is the base. The base is not changing, we just add the exponents.
      Hope this helps.
      (17 votes)
  • aqualine sapling style avatar for user Daddy-O
    Can someone explain the rule for x^a/b times y^a/b? While x^a/b times x^c/d is x^a/b plus ^c/d, x^a/b times y^a/b is somehow xy^a/b. Why are the coefficients multiplied in one case and not the other? Likewise, why are the exponents added in one case and not the other?
    (3 votes)
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    • stelly blue style avatar for user Kim Seidel
      This is based on the rules for exponents. You can do a lot more when the base is the same.
      Remember, an exponent represents repetitive multiplication of the same value: 3^4 = 3*3*3*3
      We call the "3" the base and the "4" the exponent
      When we multiply 2 items with a common base like your example with the X's, we add the exponents. For example: x^2 * x^3 = x^5
      When we multiply 2 items that do not have a common base, we are limited in what we can do.
      If the exponents happen to match like in your X and Y example, we could rewrite it as one expression raised to the same exponent. For example: x^2y^2 = (xy)^2
      Note: The parentheses are needed. If you write this as xy^2, the only item squared is the Y. Both must be squares. So, you should have (xy)^(a/b)

      It might help you if you review the properties for exponents: See this section of videos: https://www.khanacademy.org/math/in-seventh-grade-math/exponents-powers/laws-exponents-examples/v/exponent-properties-1
      (8 votes)
  • stelly green style avatar for user Marissa.L.Medina
    Should Sal have multiplied 6 times 6 in the last step to make:
    36^11/10
    instead of:
    6^11/10
    just wondering
    thanks in advance!
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      No. Exponents represent repetitive multiplication of the same value (the base). The base does not change. We just use the product property of exponents to add the exponents when we have a common base.
      For example: 6^2 * 6^3 = 6^(2+3) = 5
      You can see this makes sense if you expand the problem to not use exponents.
      6^2 * 6^3 = 6*6 * 6*6*6
      There are 5 sixes being multiplied, so the exponent becomes 5.

      Hope this helps.
      (8 votes)
  • sneak peak blue style avatar for user Stephen Earley
    Curious, why is it that when multiplying different bases with equal exponents, we treat the exponents as variables? In the case where the exponent is at all different, we’d add them together, why don’t we add them when they’re the same?
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      An exponent represent repetitive multiplication of a common value. 5^3 means 5*5*5.

      You must have a common base to combine exponents. For example: 5^3 *5^4 = 5^(3+4) = 5^7. You can see why this works if you break the problem down initially so no exponents are used.
      5^3 *5^4 = (5*5*5)*(5*5*5*5). As you can see, you are multiplying seven 5's together. So, in exponent form, it becomes 5^7

      One property of exponents says that (ab)^n = a^n*b^n. The outer exponent applies to both values inside the parentheses. For example: (2x)^3 = (2x)(2x)(2x) = (2*2*2)(x*x*x) = 2^3*x^3. The property work in both directions. So, if you start with 2^3*x^3, you can convert it to (2x)^3 it if helps you simplify an expression.

      Hope this helps.
      (7 votes)
  • hopper cool style avatar for user Atar137h
    If you can how would you simplify 6^11/10?
    (2 votes)
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    • purple pi purple style avatar for user doctorfoxphd
      You can also use the fact that x^(1+a) is the same as x¹∙x^a
      For example, we frequently simplify products of the same base by adding exponents
      3¹⁺² = 3³ = 27
      We can see that 11/10 is 1 + 1/10 so
      x^11/10 has to be the same as x¹∙x^(1/10)
      That means that 6^(11/10) = 6¹∙6^(1/10)
      so all we have to do is multiply 6 times the tenth root of 6

      You still have to find the ¹⁰√6 which is 1.96231... Multiply by the 6 and you get
      7.177387....... which is the same answer as you get by raising 6 to the 11th power and THEN taking the 10th root. At least it skips the step of having to find the 11th power.
      (4 votes)
  • mr pink red style avatar for user Lucio
    well 6^11/10 means tenth square root of 6^11 which is just 6, why didn't he simplify it completely?
    (2 votes)
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    • stelly blue style avatar for user Kim Seidel
      1) Sal's answer is fine as 6^(11/10). Radicals can be shown in their radical form or their exponential form. His answer is completely simplified for the exponential form.
      2) If you were to simplify the radical form: tenth root of 6^11 does NOT = 6. It = 6 * tenth root of 6. You changed 11/10 into 10/10 and lost 1/10 of the exponent.
      Hope this helps.
      (3 votes)

Video transcript

Let's see if we can simplify 6 to the 1/2 power times the fifth root of 6 and all of that to the third power. And I encourage you to pause this video and try it on your own. So let me actually color code these exponents, just so we can keep track of them a little better. So that's the 1/2 power in blue. This is the fifth root here in magenta. And let's see. In green, let's think about this third power. So one way to think about this fifth root is that this is the exact same thing as raising this 6 to the 1/5 power, so let's write it like that. So this part right over here, we could rewrite as 6 to the 1/5 power, and then that whole thing gets raised to the third power. And of course, we have this 6 to the 1/2 power out here, 6 to the 1/2 power times all of this business right over here. Now, what happens if we raise something to an exponent and then raise that whole thing to another exponent? Well we've already seen in our exponent properties, that's the equivalent of raising this to the product of these two exponents. So this part right over here could be rewritten as 6 to the-- 3 times 1/5 is 3/5-- 6 to the 3/5 power. And of course, we're multiplying that times 6 to the 1/2 power. 6 to the 1/2 power times 6 to the 3/5 power. And now, if you're multiplying some base to this exponent and then the same base again to another exponent, we know that this is going to be the same thing. And actually we could put these equal signs the whole way, because these all equal each other. This is the same thing as 6 being raised to the 1/2 plus 3/5 power, 1/2 plus 3 over 5. Now, what's 1/2 plus 3 over 5? Well, we could find a common denominator. It would be 10, so that's the same thing as-- actually let me just write it this way-- this is the same thing as 6 to the-- instead of 1/2, we can write it as 5/10. Plus 3/5 is the same thing as 6/10 power, which is the same thing-- and we deserve a little bit of a drum roll here, this wasn't that long of a problem-- 6 to the 11/10 power. I'll just write it all, 11/10 power. And so, that looks pretty simplified to me. I guess we're done.