Step by step
The question
Question: work out 27^(2/3).
A power that is a fraction is called a rational exponent.
It has two jobs.
The bottom number, 3, says which root: the cube root.
The top number, 2, says which power.
Why a root?
When we multiply powers of the same number, we add the powers.
27 to the power 1/3, multiplied 3 times: the powers add up to 1.
So it makes 27 to the power 1, which is 27.
That is what the cube root of 27 does.
Careful!
The power is not times.
27 times 2/3 is a different number.
There are four choices.
First we solve it, the normal way.
Then we pick.
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.
∛27 is 3, because 3 × 3 × 3 is 27.
We write 3 in place of ∛27.
2 Take the power
3 to the power 2 is 3 × 3.
That is 9.
So 27^(2/3) = 9.
3 Check
Let's check the other way: the power first, then the root.
27 to the power 2 is 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!
4 Pick the choice
Our answer is 9.
That is choice B.
5 Why not the others
A is 3.
Wrong: that is 27 to the power 1/3: the power 2 is left out.
C is 18.
Wrong: 27 × 2/3: multiplied by the power.
That is the Careful mistake from before.
D is 729.
Wrong: 27 × 27: forgot the cube root.
Let's look back.
First: take the root.
Then: take the power.
Then: check.
Then: pick the choice.
Last: why not the others.