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## Algebra 2

### Course: 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

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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.

## Want to join the conversation?

- 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?(9 votes)
- That is a very good question. When simplifying a radical, the quickest method (besides intuition) is to make a factor tree. A factor tree is something you may have learned about in grade school. Here is a quick refresher:
`https://www.khanacademy.org/math/in-sixth-grade-math/playing-numbers/prime-factorization/v/prime-factorization`

The next step would be to try and find number combinations that go in the same amount of times as the index. This video explains how to do that:`https://www.khanacademy.org/math/algebra/rational-exponents-and-radicals/introduction-to-rational-exponents-and-radicals/v/radical-expressions-with-higher-roots`

I hope this helps! Good luck!(20 votes)

- At1:38Sal 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!(14 votes)
- It does end up with the same result to take the fraction and put it as a denominator under a one as the numerator, however it is faster and less confusing to do it the way Sal does in the example.(8 votes)

- At0:45, 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?(9 votes)
- Yes, it should be in parentheses. He said the correct thing: "5/3 squared" but he sashayed on to the next example without putting in that clarifying set of parentheses.(7 votes)

- Couldn't you also find the prime factorization to find 256^1/4 or 81^1/4?(5 votes)
- Yes, that is the way that you would do it without a calculator. You could also just remember 4^4 is 256 and 3^4 is 81.(6 votes)

- I can't find any videos on how to do 1/2 with an exponent of 2(3 votes)
- Remember, an exponent tells you to multiply the base (1/2) by the number of times in the exponents.

(1/2)^2 = (1/2)(1/2) = (1*1)/(2*2) = 1/4

Here's the video: https://www.khanacademy.org/math/cc-sixth-grade-math/x0267d782:cc-6th-exponents-and-order-of-operations/x0267d782:powers-of-fractions-and-decimals/v/powers-of-fractions

FYI - You can find topics by using the search bar at the top of any KhanAcademy page. That's how I found the video.(3 votes)

- What if you had a decimal number to the power of a negative exponent? for example 0.3 ^ -4(3 votes)
- 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(3 votes)

- How would I simplify a problem that is to the fourth root, but has a negative base? Something like (-625)^(3/4).(3 votes)
- 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.(3 votes)

- if it the problem is (125/27) ^ -2/3 how would you solve it?(3 votes)
- (125/27)^(-2/3)=1/(125/27)^(2/3)=1/[(125/27)^(1/3)]^2=1/[(5/3)^2]=1/(25/9)=9/25.(3 votes)

- why are you only looking for the principle root?(3 votes)
- The negative roots are unnecessary for videos on negative fractional exponents.(2 votes)

- At1:30Sal 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)
- 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.(5 votes)

## 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.