Step by step
The question
Question: work out 1000^(2/3).
1000^(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.
1000 times 2/3 is a different number.
Careful: not 1000^(2/3) = 1000 × 23 ✗
1 Take the root
The bottom is 3: the cube root of 1000.
The top is 2: to the power 2.
We take the root first.
It keeps the numbers small.
= (3√1000)2
∛1000 is 10, because 10 × 10 × 10 is 1000.
10 × 10 × 10 = 1000
We write 10 in place of ∛1000.
= 102
2 Take the power
10 to the power 2 is 10 × 10.
That is 100.
10 × 10 = 100
So 1000^(2/3) = 100.
= 100
3 Check
Let's check the other way: the power first, then the root.
1000 to the power 2 is 1000000.
Other order: 10002 = 1000000
Let's look back.
First: take the root.
Then: take the power.
Last: check.
Answer
= 100