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### Course: Integrated math 2 > Unit 6

Lesson 4: Evaluating exponents and 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

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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.

## Want to join the conversation?

- 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)
- because when you multiply the same bases you just add the exponents(15 votes)

- I am in intermediate algebra/trig in COLLEGE and this is what I am learning; how is this Algebra I ? :((10 votes)
- Well don't feel bad I learned this in 8 th grade and sometimes they revisit topics you have already visited before again :)(14 votes)

- How would you do this with different bases instead?(19 votes)
- You would have to manually calculate them (if it was multiplication).(1 vote)

- 1:52, when we multiply 6^1/2 and 6^3/5 together, wouldn't that equal 36^11/10?(6 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!(19 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(6 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.(17 votes)

- 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)- 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.(10 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?(3 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(8 votes)

- 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)
- 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)

- 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)

- well 6^11/10 means tenth square root of 6^11 which is just 6, why didn't he simplify it completely?(2 votes)
- 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.