Step by step
The question
Question: simplify ∛40.
The small 3 says how many times.
∛40 asks: which number, times itself 3 times, makes 40?
1 Find the biggest cube
Cubes are a number times itself 3 times.
2 × 2 × 2 is 8.
We try the biggest first.
Does 8 go into 40?
Yes: 40 divided by 8 is 5.
Careful!
4 is a square, not a cube.
The cube root of 4 is not 2.
We need a number times itself 3 times.
2 Split the root
So 40 is 8 times 5.
We write it under the root.
Rule: a root over a times can be split into two roots.
So it is ∛8 times ∛5.
∛8 is 2, because 2 × 2 × 2 is 8.
We write 2 in place of ∛8.
We leave out the times sign.
5 has no cube in it, so we are done.
So ∛40 = 2∛5.
3 Check
Let's check: we take the answer to the power 3.
∛5 to the power 3 is just 5.
2 × 2 × 2 is 8.
Times 5 is 40.
We get 40 back.
So the answer is right!
Let's look back.
First: find the biggest cube.
Then: split the root.
Last: check.