Step by step
The question
Question: work out 81^(−3/4).
A fraction power has two jobs.
The bottom number, 4, says which root: the 4th root.
The top number, 3, says which power.
Careful!
A minus in the power does not make the answer negative.
It means 1 over.
1 Remove the minus
A minus power means 1 over the same power without the minus.
So 81^(−3/4) is 1/81^(3/4).
2 Take the root
The bottom is 4: the 4th root of 81.
The top is 3: to the power 3.
We take the root first.
It keeps the numbers small.
The 4th root of 81 is 3, because 3 × 3 × 3 × 3 is 81.
We write 3 in place of the 4th root of 81.
3 Take the power
3 to the power 3 is 3 × 3 × 3.
That is 27.
So 81^(−3/4) = 1/27.
4 Check
Let's check.
Without the minus, 81^(3/4) is 27.
A power times its minus power is 81 to the power 0, which is 1.
27 times 1/27 is 1.
So the answer is right!
Let's look back.
First: remove the minus.
Then: take the root.
Then: take the power.
Last: check.