Step by step
The question
Question: work out 32^(3/5).
A fraction power has two jobs.
The bottom number, 5, says which root: the 5th root.
The top number, 3, says which power.
Careful!
The power is not times.
32 times 3/5 is a different number.
1 Take the root
The bottom is 5: the 5th root of 32.
The top is 3: to the power 3.
We take the root first.
It keeps the numbers small.
The 5th root of 32 is 2, because 2 × 2 × 2 × 2 × 2 is 32.
We write 2 in place of the 5th root of 32.
2 Take the power
2 to the power 3 is 2 × 2 × 2.
That is 8.
So 32^(3/5) = 8.
3 Check
Let's check the other way: the power first, then the root.
32 to the power 3 is 32768.
The 5th root of 32768 is 8, because 8 × 8 × 8 × 8 × 8 is 32768.
We get 8 again.
But the numbers were bigger.
So root first is easier.
And the answer is right!
Let's look back.
First: take the root.
Then: take the power.
Last: check.