Step by step
The question
Question: work out 8^(2/3).
8^(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.
a^(m/n) = (n√a)m
Careful!
The power is not times.
8 times 2/3 is a different number.
Careful: not 8^(2/3) = 8 × 23 ✗
1 Take the root
The bottom is 3: the cube root of 8.
The top is 2: to the power 2.
We take the root first.
It keeps the numbers small.
= (3√8)2
∛8 is 2, because 2 × 2 × 2 is 8.
2 × 2 × 2 = 8
We write 2 in place of ∛8.
= 22
2 Take the power
2 to the power 2 is 2 × 2.
That is 4.
2 × 2 = 4
So 8^(2/3) = 4.
= 4
3 Check
Let's check the other way: the power first, then the root.
8 to the power 2 is 64.
Other order: 82 = 64
The cube root of 64 is 4, because 4 × 4 × 4 is 64.
We get 4 again.
But the numbers were bigger.
So root first is easier.
And the answer is right!
3√64 = 4, 4 × 4 × 4 = 64
Let's look back.
First: take the root.
Then: take the power.
Last: check.
Answer
= 4