Step by step
The question
Question: simplify ∛81.
The small 3 says how many times.
∛81 asks: which number, times itself 3 times, makes 81?
1 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 81?
Yes: 81 divided by 27 is 3.
Careful!
81 is a square, not a cube.
The cube root of 81 is not 9.
We need a number times itself 3 times.
2 Split the root
So 81 is 27 times 3.
We write it under the root.
Rule: a root over a times can be split into two roots.
So it is ∛27 times ∛3.
∛27 is 3, because 3 × 3 × 3 is 27.
We write 3 in place of ∛27.
We leave out the times sign.
3 has no cube in it, so we are done.
So ∛81 = 3∛3.
3 Check
Let's check: we take the answer to the power 3.
∛3 to the power 3 is just 3.
3 × 3 × 3 is 27.
Times 3 is 81.
We get 81 back.
So the answer is right!
Let's look back.
First: find the biggest cube.
Then: split the root.
Last: check.