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Fractional Roots

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Fractional Roots

by gmatstinks123 » Sat Mar 03, 2012 4:43 am
I am a bit confused at the concept of fractional exponents. I understand that 8 ^ (2/3) is equal to the third root of (8^2) = 4; another way of doing this is taking the third root of 8 then taking that total and squaring it = 4.

Using the same approaches used above why does √[(-5) ^2] = 5 and (√-5) ^ 2 = undefined?

Why are there contradictory results here when we are using basic properties of fractional exponents?

Also, how is √[(-5)^2] = 5? Shouldn't it be equal to (-5) ^ (2/2) = -5? I used (2/2) because the exponent that the base is raised to is 2, and the root is 2.

Last but not least, why is (-5) ^ (2/2) = -5? Using the properties why can't it be equal to √[(-5) ^2] = 5, or (√-5) ^ 2 = undefined?

I know some of these questions may seem redundant but I am just trying to master this concept. Thank you guys so much for your feedback!
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by vikram4689 » Sat Mar 03, 2012 11:14 am
Not some of these BUT all of these is redundant. You are trespassing into the ares of calculus (functions) where we have calculate limits on the variables for which these properties hold. These properties does not exist for domain -infinity to +infinity.
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by krusta80 » Sat Mar 03, 2012 5:14 pm
gmatstinks123 wrote:I am a bit confused at the concept of fractional exponents. I understand that 8 ^ (2/3) is equal to the third root of (8^2) = 4; another way of doing this is taking the third root of 8 then taking that total and squaring it = 4.

Using the same approaches used above why does √[(-5) ^2] = 5 and (√-5) ^ 2 = undefined?

Why are there contradictory results here when we are using basic properties of fractional exponents?

Also, how is √[(-5)^2] = 5? Shouldn't it be equal to (-5) ^ (2/2) = -5? I used (2/2) because the exponent that the base is raised to is 2, and the root is 2.

Last but not least, why is (-5) ^ (2/2) = -5? Using the properties why can't it be equal to √[(-5) ^2] = 5, or (√-5) ^ 2 = undefined?

I know some of these questions may seem redundant but I am just trying to master this concept. Thank you guys so much for your feedback!
This is really just a matter of order of operations.
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by ceilidh.erickson » Fri Dec 07, 2012 8:14 am
krusta80 is right, this is a question of order of operations. With (√-5) ^ 2 , you have to start inside the parentheses, and you can't take the root of a negative. With √[(-5)^2] = 5 , start with squaring -5, and you get 25.

Here's where it gets tricky: the GMAT says that when a root sign is given, as in x = √25, the correct answer is only the positive root: 5.

If there is an exponent rather than a root, then the answer is positive or negative: x^2 = 25 ... x = 5 or -5

The GMAT will use this concept this way: What is √(x^2) ? It should just be x, right? We're squaring, then square rooting. The problem is that squaring obscures the sign of x, and when we take the root, we only get the positive value. So if x is positive, √(x^2) = x. If x is negative, √(x^2) = -x.

√(x^2) is really just another way of saying |x|.
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Harvard Graduate School of Education
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