SOLUTION: How would you evaluate an expression with a mixed-number exponent such as 8^1 1/3?

Question 374185: How would you evaluate an expression with a mixed-number exponent such as 8^1 1/3?
Found 2 solutions by Fombitz, jsmallt9:
Answer by Fombitz(32388) (Show Source): You can put this solution on YOUR website!
DUPLICATE See Square-cubic-other-roots/374194:
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

Mixed numbers are almost always a pain. I usually recommend that they be converted to improper fractions. It is the same here. So we will start by changing to 4/3:

If you are still not sure how to simplify this, I recommend writing the exponent in a factored form. In this case, we would rewrite the expression as:

Since

which says that we raise 8 to the 4th power and then, because an exponent of 1/3 means cube root, find the cube root of the answer.
Also, since multiplication is Commutative, we are free to change the order:

which says to find a cube root and then raise to the 4th power.

So, in summary, an exponent of means we will raise to the 4th power and we will find a cube root and we get to choose the order in which these are done!

Since 8 is a perfect cube (), it seems to me that finding the cube root first will be easier than raising 8 to the 4th power. So here's the problem from start to finish:

(Note: even if you decide to raise to the 4th power first and then find the cube root, you still end up with 16.)
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