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## College Algebra

### Course: College Algebra>Unit 13

Lesson 3: Evaluating exponents and radicals

# Evaluating fractional exponents: negative unit-fraction

How to evaluate powers that are negative unit fractions, like 9 raised to -½ and 27 raised to -⅓. Created by Sal Khan.

## Want to join the conversation?

• Why do we invert the number (reciprocal) if there's a negative exponent? Is there any logical explanation?
• When a number is raised to a negative exponent, we invert the number to make the exponent positive. Its basically a negative number to a negative exponent makes a positive number and exponent. (yes i know im 8 years late, but hope this helps)
• how do you find the answer to a fraction ^ to a fraction i don't know how to do that
• (1/2)^2 = (1/2 x 1/2) = 1/4

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

(1/4)^(1/2) = the square root of 1/4

The square root of 1/4 is 1/2, since (1/2)^2 = 1/4

If you get something like (5/9)^(1/2), take the square root of the numerator and denominator separately. Here the square root of 5 is irrational and can be left as "the square root of 5," However, the square root of 9 = 3.

• is it right to say that any negative number to the power of an even number is undefined, but a negative number to the power of an odd number has a solution?
• Any negative number raised to the power of an even number ALWAYS results in a positive number,
eg (-1)²=1, or (-2)²=4.
Any negative number raised to a power of an odd number ALWAYS results in a negative number,
eg (-1)³=-1, or (-2)³=-8.
• in this example (-27)^-1/3 --> 1/(-27)^1/3

isn't this also the root of -27 cubed? And if so how come the negative sqrt here doesn't make this a "no solution"
• The -1/3 exponent means take the third root of the reciprocal. So remember that any number when divided by 1 is equal to the number itself. The negative exponent means take the reciprocal, or flip the fraction, so,
( (-27)^-1/3) / 1 = 1 / ( (-27)^1/3), and the negative exponent is now a positive exponent. Regarding the fractional exponent, if the expression were telling you to cube, then the 3 would be in the numerator, but the 3 is in the denominator, so, you are supposed to take the third root, or cubed root. So, the expression, simplified, equals, 1/-3, or - 1/3, because (-3) * (-3) * (-3) = -27. Also, later, you will learn that there are solutions to negative square roots. Hope that helps, and good luck in your studies!
• In the video at (-27)^-1/3 is equal to 1/(-27)^1/3. I understand the exponent changing signing to a positive when it is flipped. But why doesn't the (-27) not change signs when it is flipped?

• The properties of exponents don't change the base. They just work with the exponents. In this case, the property being used is the one that converts a negative exponent to a positive. It tells us that we can do this by using the reciprocal of the base. The reciprocal of (-27)^(-1/3) = 1/(-27)^(1/3)

Note: the reciprocal of any number will carry the same sign as the original. For example: the reciprocal of -3/4 = -4/3

Hope this helps.
• Yay i got the question right without watching the walkthrough
• This is seems like a stupid question to me, but I haven't really given much thought into it until now. Do imaginary numbers only apply to even numbered radicals? I am asking this because in , Sal does not use any imaginary numbers.
• You are correct, the reason is that any number to an odd power is negative, so you can take cubed root, fifth root, etc. of negative numbers. For example the cubed root of -8 is -2 because -2 * -2 * -2 = -8.
• How do you work out a fraction to the power of a negative fraction?
• `(a/b)^-(n/m) = 1/(a/b)^(n/m) = 1/((a/b)^(1/m))^n`
`(a/b)^(1/m)` is an m-th root of a/b, for example `(4/9)^(1/2) = √(4/9) = 2/3`. So you end up with:
`1/(m-root(a/b))^n`

Ex:
`(16/25)^-(3/2) = 1/(16/25)^(3/2) = 1/((16/25)^(1/2))^3`
`= 1/(√(16/25))^3 = 1/(4/5)^3 = 1/(64/125) = 125/64`
• How would you solve something that the entire exponent is negative? For example if you have a equation like - 12/x^3, just to demonstrate it's the entire part that's negative so -(12/x^3) would be equivalent. Could you just flip the whole thing and make it x^3/12 or is there something I'm missing?
• When you flip something and change its sign, you change it.
Think of -1/2. That's not the same as 2/1.
But you are correct that -12/x^3 and -(12/x^3) are equivalent.
You must separate in your mind the difference between a negative sign for a number, and a negative exponent.
For example, 2^-3 = 1/2^3. These are both equal to 1/8, which is positive.

I hope this helps clarify things for you.