Step by step
The question
Question: simplify ∛−54.
The small 3 says how many times.
∛−54 asks: which number, times itself 3 times, makes −54?
1 Look at the sign
3 is odd.
A minus number times itself 3 times stays minus.
So the answer is minus.
So ∛−54 is minus ∛54.
2 Find the biggest cube
Cubes are a number times itself 3 times.
2 × 2 × 2 is 8.
3 × 3 × 3 is 27.
We try the biggest first.
Does 27 go into 54?
Yes: 54 divided by 27 is 2.
Careful!
9 is a square, not a cube.
The cube root of 9 is not 3.
We need a number times itself 3 times.
3 Split the root
So 54 is 27 times 2.
We write it under the root.
Rule: a root over a times can be split into two roots.
So it is ∛27 times ∛2.
∛27 is 3, because 3 × 3 × 3 is 27.
We write 3 in place of ∛27.
We leave out the times sign.
2 has no cube in it, so we are done.
So ∛−54 = −3∛2.
4 Check
Let's check: we take the answer to the power 3.
∛2 to the power 3 is just 2.
(−3) × (−3) × (−3) is −27.
Times 2 is −54.
We get −54 back.
So the answer is right!
Let's look back.
First: look at the sign.
Then: find the biggest cube.
Then: split the root.
Last: check.