How to deal with fractional exponents: A^(M/N) = Nsqrt(A^M) = (Nsqrt(A)^M

[Ted_Grendy]

Hello all

I wanted to ask about how to deal with fractional exponents.

Would I be correct in saying that the following statement are all equal:-

A^(M/N) = Nsqrt(A^M) = (Nsqrt(A)^M

In the left I have A as the base and I am raising it to the power M/N.
In the middle I have the Nthroot of A to the power M.
To the right I have Nthroot of A and the result being raised to the power M.

Will these ALWAYs hold true?

Thank you.

 

[Steven G]

Hello all

I wanted to ask about how to deal with fractional exponents.

Would I be correct in saying that the following statement are all equal:-

A^(M/N) = Nsqrt(A^M) = (Nsqrt(A)^M

In the left I have A as the base and I am raising it to the power M/N.
In the middle I have the Nthroot of A to the power M.
To the right I have Nthroot of A and the result being raised to the power M.

Will these ALWAYs hold true?

Thank you.

Consider the following problem: (-9)^2/2.
1) (-9)^2/2 = (sqrt(-9))² = (3i)² = --9 (note: if you did not learn how to take square roots of neg numbers, then you can't even compute (sqrt(-9))²!!!
2) (-9)^2/2 = sqrt((-9)²) = sqrt(81) = 9
Note that we did not get the same answers!!

 

[Dr.Peterson]

Hello all

I wanted to ask about how to deal with fractional exponents.

Would I be correct in saying that the following statement are all equal:-

A^(M/N) = Nsqrt(A^M) = (Nsqrt(A)^M

In the left I have A as the base and I am raising it to the power M/N.
In the middle I have the Nthroot of A to the power M.
To the right I have Nthroot of A and the result being raised to the power M.

Will these ALWAYs hold true?

Thank you.

The answer is: Yes, but ...

They are true for positive A, and as long as you are only considering positive real roots.

But ... once you allow negative or complex roots, you have to allow multiple roots (that is, you can't just stick with "principal roots"). And then you don't have a function in the usual sense, so it becomes questionable whether two expressions can be "equal" at all.

However, for ordinary use, these are valid, and worth knowing. (But don't call it "Nsqrt"; the nth root is not the "nth square root", as "square" refers only to the "2nd root".)