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### Course: Algebra 2>Unit 6

Lesson 3: Evaluating exponents & radicals

# 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?

• At Sal 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!
• 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.
• 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?
• 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!
• At , 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?
• 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.
• I can't find any videos on how to do 1/2 with an exponent of 2
• Couldn't you also find the prime factorization to find 256^1/4 or 81^1/4?
• 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.
• How would I simplify a problem that is to the fourth root, but has a negative base? Something like (-625)^(3/4).
• 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

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.
• At Sal 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?
• 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.
• What if you had a decimal number to the power of a negative exponent? for example 0.3 ^ -4
• 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
• if it the problem is (125/27) ^ -2/3 how would you solve it?
• (125/27)^(-2/3)=1/(125/27)^(2/3)=1/[(125/27)^(1/3)]^2=1/[(5/3)^2]=1/(25/9)=9/25.