How do I determine which of these expressions is greater WITHOUT evaluating them? (Math, Algebra, Exponents)?

Well there are several way to approach this, but it depends what you are best at. First off with fractional exponenets they might look scary but they are simple. Really -2/5 is just 5 root 1/32 to the negative second power.

The top of the fraction or the numerator is to the power of, while the denominator is the root of. An easy example would be like 4 ^ 1/2 that is just 4 to the first power root 2 or just the square root of 2 Second negative exponents: so an number to the power of -1 is just that number becomes a denominator and the numerator becomes a 1 for example 3 ^ -1 is 1/3. With negative exponents you put a 1 as the numerator and then the orignal number is to the power of the negative exponent.

For example lets use 4 to the negative second power 4^-2 that would just be 1/(4)^2 or 1/16. The denominator 4 is solved by applying the power of 2 to it. So to put that all together here is the solution to your problem It all depends how you want to approach but this is how I approached it.

First I expanded the problem 1/square root of 32 is also the square root of 32 to the power of the -1 ((((square root of 32)^-1)^-2)^1/5) The square root of 32 can also be written as 32 to the 1/2 power or (((((32)^(1/2))^-1)^-2)^1/5) With power of exponent properties you can simplify this (1/2) multiplied by -1 the multiplied by -2 then multiplied by 1/5 First:1/2*-1= -1/2 Second:-1/2*-2=+1 Third:1*1/5=1/5 So after doing all that you get (32)^(1/5) Or 32 to the 1/5 power also known as the root 5 of 32.

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