The question
Question: find ∛27.
3√27
The small 3 says how many times.
∛27 asks: which number, times itself 3 times, makes 27?
? × ? × ? = 27
1 List the cubes
Cubes are a number times itself 3 times.
2 × 2 × 2 is 8.
3 × 3 × 3 is 27.
Cubes: 23 = 8, 33 = 27
Careful!
The cube root is not 27 divided by 3.
9 × 9 × 9 is far more than 27.
Careful: not 3√27 = 9 ✗
27 is in the list: 3 × 3 × 3 is 27.
3 × 3 × 3 = 27
So ∛27 = 3.
= 3
2 Check
Let's check: 3 times itself 3 times.
We get 27.
So the answer is right!
3 × 3 × 3 = 27
Let's look back.
First: list the cubes.
Last: check.
▶ Watch it solved
The question
Question: find ∛−8.
3√−8
The small 3 says how many times.
∛−8 asks: which number, times itself 3 times, makes −8?
? × ? × ? = −8
1 Look at the sign
3 is odd.
A minus number times itself 3 times stays minus.
So the answer is minus.
(−) × (−) × (−) = −
So ∛−8 is minus ∛8.
= −3√8
2 List the cubes
Cubes are a number times itself 3 times.
2 × 2 × 2 is 8.
3 × 3 × 3 is 27.
Cubes: 23 = 8, 33 = 27
8 is in the list: 2 × 2 × 2 is 8.
2 × 2 × 2 = 8
So ∛−8 = −2.
= −2
3 Check
Let's check: −2 times itself 3 times.
We get −8.
So the answer is right!
(−2) × (−2) × (−2) = −8
Let's look back.
First: look at the sign.
Then: list the cubes.
Last: check.
▶ Watch it solved
The question
Question: simplify ∛−54.
3√−54
The small 3 says how many times.
∛−54 asks: which number, times itself 3 times, makes −54?
? × ? × ? = −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.
= −3√54
2 Find the biggest cube
Cubes are a number times itself 3 times.
2 × 2 × 2 is 8.
3 × 3 × 3 is 27.
Cubes: 23 = 8, 33 = 27
We try the biggest first.
Does 27 go into 54?
Yes: 54 divided by 27 is 2.
54 ÷ 27 = 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.
Careful: not 3√9 × 6 = −33√6 ✗
3 Split the root
So 54 is 27 times 2.
We write it under the root.
= −3√27 × 2
Rule: a root over a times can be split into two roots.
3√a × b = 3√a × 3√b
So it is ∛27 times ∛2.
= −3√27 × 3√2
∛27 is 3, because 3 × 3 × 3 is 27.
3 × 3 × 3 = 27
We write 3 in place of ∛27.
= −3 × 3√2
We leave out the times sign.
2 has no cube in it, so we are done.
So ∛−54 = −3∛2.
= −33√2
4 Check
Let's check: we take the answer to the power 3.
∛2 to the power 3 is just 2.
(−33√2)3 = (−3) × (−3) × (−3) × 2
(−3) × (−3) × (−3) is −27.
Times 2 is −54.
We get −54 back.
So the answer is right!
= −27 × 2 = −54
Let's look back.
First: look at the sign.
Then: find the biggest cube.
Then: split the root.
Last: check.
▶ Watch it solved
