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Course: Integrated math 2>Unit 6

Lesson 4: 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.

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• 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!
• tip of the day: if you are on mac, try pressing Option + V. It will output the radical sign (√)
• 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.
• 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 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.
• what if you raise (-3) to the 1/2 power (I know it's undefined but why is it undefined)
• Raising a number to the ½ power is the same as finding its square root. If we restrict ourselves to the real numbers then the square root of -3 (or any negative number) does not exist.
• 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`

Video transcript

Let's do some slightly more complicated fractional exponent examples. So we already know that if I were to take 9 to the 1/2 power, this is going to be equal to 3, and we know that because 3 times 3 is equal to 9. This is equivalent to saying, what is the principal root of 9? Well, that is equal to 3. But what would happen if I took 9 to the negative 1/2 power? Now we have a negative fractional exponent, and the key to this is to just not get too worried or intimidated by this, but just think about it step by step. Just ignore for the second that this is a fraction, and just look at this negative first. Just breathe slowly, and realize, OK, I got a negative exponent. That means that this is just going to be 1 over 9 to the 1/2. That's what that negative is a cue for. This is 1 over 9 to the 1/2, and we know that 9 to the 1/2 is equal to 3. So this is just going to be equal to 1/3. Let's take things a little bit further. What would this evaluate to? And I encourage you to pause the video after trying it, or pause the video to try it. Negative 27 to the negative 1/3 power. So I encourage you to pause the video and think about what this would evaluate to. So remember, just take a deep breath. You can always get rid of this negative in the exponent by taking the reciprocal and raising it to the positive. So this is going to be equal to 1 over negative 27 to the positive 1/3 power. And I know what you're saying. Hey, I still can't breathe easily. I have this negative number to this fractional exponent. But this is just saying what number, if I were to multiply it three times-- so if I have that number, so whatever the number this is, if I were to multiply it, if I took three of them and I multiply them together, if I multiplied 1 by that number three times, what number would I have to use here to get negative 27? Well, we already know that 3 to the third, which is equal to 3 times 3 times 3, is equal to positive 27. So that's a pretty good clue. What would negative 3 to the third power be? Well, that's negative 3 times negative 3 times negative 3, which is negative 3 times negative 3 is positive 9. Times negative 3 is negative 27. So we've just found this number, this question mark. Negative 3 times negative 3 times negative 3 is equal to negative 27. So negative 27 to the 1/3-- this part right over here-- is equal to negative 3. So this is going to be equal to 1 over negative 3, which is the same thing as negative 1/3.