Step by step
The question
Question: work out 125^(2/3).
A fraction power has two jobs.
The bottom number, 3, says which root: the cube root.
The top number, 2, says which power.
Careful!
The power is not times.
125 times 2/3 is a different number.
1 Take the root
The bottom is 3: the cube root of 125.
The top is 2: to the power 2.
We take the root first.
It keeps the numbers small.
∛125 is 5, because 5 × 5 × 5 is 125.
We write 5 in place of ∛125.
2 Take the power
5 to the power 2 is 5 × 5.
That is 25.
So 125^(2/3) = 25.
3 Check
Let's check the other way: the power first, then the root.
125 to the power 2 is 15625.
The cube root of 15625 is 25, because 25 × 25 × 25 is 15625.
We get 25 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.