Step by step
The question
Question: work out 64^(2/3).
64^(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.
64 times 2/3 is a different number.
Careful: not 64^(2/3) = 64 × 23 ✗
1 Take the root
The bottom is 3: the cube root of 64.
The top is 2: to the power 2.
We take the root first.
It keeps the numbers small.
= (3√64)2
∛64 is 4, because 4 × 4 × 4 is 64.
4 × 4 × 4 = 64
We write 4 in place of ∛64.
= 42
2 Take the power
4 to the power 2 is 4 × 4.
That is 16.
4 × 4 = 16
So 64^(2/3) = 16.
= 16
3 Check
Let's check the other way: the power first, then the root.
64 to the power 2 is 4096.
Other order: 642 = 4096
The cube root of 4096 is 16, because 16 × 16 × 16 is 4096.
We get 16 again.
But the numbers were bigger.
So root first is easier.
And the answer is right!
3√4096 = 16, 16 × 16 × 16 = 4096
Let's look back.
First: take the root.
Then: take the power.
Last: check.
Answer
= 16