Cubes and Cube roots


1. What is a Cube?

A number multiplied by itself three times results in a cube.

A cube is equal to a number raised to power 3.

2 × 2 × 2 = 23 = 8

8 is a cube equal to 2 raised to power 3.

Similarly, 27/64 = (3/4)3

2. Perfect Cube:

The cube of a natural number is called a Perfect Cube.

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 are perfect cubes of the first 10 natural numbers respectively.

3. How to Check if a Number is a Perfect Cube?

Resolve the number into its prime factors. If the powers in the factors are multiples of 3, then the number is a Perfect Cube.

Examples:

1. Is 1728 a perfect cube?

Solution:

Resolve 1728 into its prime factors.

1728 = 33 × 26.

Since both the powers 3 and 6 are multiples of 3, 1728 is therefore a perfect cube.

2. Is 648 a perfect cube?

Solution:

648 = 23 × 34

In 34, the power 4 is not a multiple of 3.

So, 648 is not a perfect cube.

3. Cube Root:

The cube root of a number N is a number a, which results in N on being multiplied with itself three times.

i.e. if a × a × a = N, i.e. a3 = N, then a = 3√N

a is said to be the cube root of N.

The cube root of a number N is denoted as 3√N

3√ is the symbol to denote cube root. In this, the index is 3.

Note:

The symbol for square root 2√ can be also written as √ by dropping the index 2.

But the index 3 in the cube root symbol 3√ has to be shown.

4. How to find the cube root of a number?

Step 1:

Resolve the given number into its prime factors

Step 2:

Divide the powers of the prime factors with 3

Step 3:

The cube root is now the product of the prime factors with quotients obtained in step 2 written as the powers (of the prime factors)

Solved Examples:

1. What is the cube root of 8?

Solution:

8 = 2 × 2 × 2 = 23

Divide the power 3 by 3. The quotient is 1.

Now the cube root of 8 is 21, i.e. 2.

2. What is the cube root of 216?

Solution:

216 = 23 × 33

Divide each of the powers 3 by 3. The quotients are 1 each.

Now the cube root is:

21 × 31 = 2 × 3 = 6