Step by step
The question
Question: work out 27^(2/3).
27^(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.
27 times 2/3 is a different number.
Careful: not 27^(2/3) = 27 × 23 ✗
1 Take the root
The bottom is 3: the cube root of 27.
The top is 2: to the power 2.
We take the root first.
It keeps the numbers small.
= (3√27)2
∛27 is 3, because 3 × 3 × 3 is 27.
3 × 3 × 3 = 27
We write 3 in place of ∛27.
= 32
2 Take the power
3 to the power 2 is 3 × 3.
That is 9.
3 × 3 = 9
So 27^(2/3) = 9.
= 9
3 Check
Let's check the other way: the power first, then the root.
27 to the power 2 is 729.
Other order: 272 = 729
The cube root of 729 is 9, because 9 × 9 × 9 is 729.
We get 9 again.
But the numbers were bigger.
So root first is easier.
And the answer is right!
3√729 = 9, 9 × 9 × 9 = 729
Let's look back.
First: take the root.
Then: take the power.
Last: check.
Answer
= 9