Question

Use the rules for raising a power to a power to explain why raising a number to the power of ½ is the same as finding the square root of the number.

Answer 1

When you multiply a variable times itself, the exponents get added up.
\[(x^4)(x^4)=x^8\]
When you want the square root of a number, you want to know the exponent that your variable will have so that when it gets multiplied times itself will yield the current exponent.
Since it will be multiplying itself, the exponent will added times 2. So the way to get the square root is to divide the exponent in half. or *MULTIPLIED TIMES 1/2*
\[\sqrt{x^8}=(X^{(8)})^{\frac{1}{2}} \]
or raise to 1/2 the current exponent.
Because when you raise to a power, you multiply times the exponent.
This comes out to 8*(1/2)=4

My explanation is not very clear. I'M having trouble following it.
I hope this helps you by pointing you in the right direction, but you'll have to come up with better wording than I've done here.
Good luck!

FYI- This works for higher roots too. raising to 1/3 gets you the cubed root, etc.