Step by step
The question
Question: simplify ∛128.
The small 3 says how many times.
∛128 asks: which number, times itself 3 times, makes 128?
1 Find the biggest cube
Cubes are a number times itself 3 times.
2 × 2 × 2 is 8.
3 × 3 × 3 is 27.
4 × 4 × 4 is 64.
We try the biggest first.
Does 64 go into 128?
Yes: 128 divided by 64 is 2.
Careful!
16 is a square, not a cube.
The cube root of 16 is not 4.
We need a number times itself 3 times.
2 Split the root
So 128 is 64 times 2.
We write it under the root.
Rule: a root over a times can be split into two roots.
So it is ∛64 times ∛2.
∛64 is 4, because 4 × 4 × 4 is 64.
We write 4 in place of ∛64.
We leave out the times sign.
2 has no cube in it, so we are done.
So ∛128 = 4∛2.
3 Check
Let's check: we take the answer to the power 3.
∛2 to the power 3 is just 2.
4 × 4 × 4 is 64.
Times 2 is 128.
We get 128 back.
So the answer is right!
Let's look back.
First: find the biggest cube.
Then: split the root.
Last: check.