Main content

## Algebra 2

### Unit 6: Lesson 3

Evaluating exponents & radicals- Evaluating fractional exponents
- Evaluating fractional exponents: negative unit-fraction
- Evaluating fractional exponents: fractional base
- Evaluating quotient of fractional exponents
- Evaluating mixed radicals and exponents
- Evaluate radical expressions challenge

© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

# Evaluating mixed radicals and exponents

CCSS.Math: ,

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.

## Want to join the conversation?

- How would you do this with different bases instead?(15 votes)
- I am in intermediate algebra/trig in COLLEGE and this is what I am learning; how is this Algebra I ? :((7 votes)
- I'm in grade 10 and we're doing this. :/(4 votes)

- 1:52, when we multiply 6^1/2 and 6^3/5 together, wouldn't that equal 36^11/10?(4 votes)
- No, whenever you are multiplying exponents with the same bases, you always keep the base and add the exponents. (a)^b x (a)^c = a^b+c

So in this case, 6^1/2 x 6^3/5 = 6^1 1/10

I hope this helped!(13 votes)

- Shouldn't the answer be 36^11/10 because 6*6=36 and not 6? If not then how did he get 6?(2 votes)
- because when you multiply the same bases you just add the exponents(6 votes)

- 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(2 votes)- 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.(8 votes)

- If you can how would you simplify 6^11/10?(2 votes)
- 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)

- 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?(2 votes)
- 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(5 votes)

- How would you simplify radicals with exponents? Ex: ³√216(3 votes)
- This simply means 216^1/3, then you can go from there

Sorry if I'm a little late, but I hope this helped!(1 vote)

- could i have some help with this please

(2/3)-3 (-3 it the power)

thanks(1 vote)- You distribute exponents to every number in the term that's multiplied/divided, so your expression becomes this:

(2/3)^(-3) = (2^(-3)) / (3^(-3))

Now, you know that a negative exponent is the reciprocal of the positive version. This allows you to make negative exponents in the denominator positive exponents in the numerator, and vice versa.

= 3^3 / 2^3

= 27 / 8(3 votes)

- in next Practice problem, we are asked to Evaluate -100 under fifth root sign. how this can be possible since can take only root of 5 from positive number?(1 vote)
- If the index is odd (5th root) you can do negative numbers. This is because if you raise a negative number to an odd power, you get a negative result. For example:

(-2)^3 = -8; so cubert(-8) = -2

(-2)^5 = -32; so 5throot(-32) = -2

If the index of the radical is even (square root, 4th root, etc), then we can't do negative numbers within the real number system.

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.