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 = 2^{3} = 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 = 3^{3} × 2^{6}.

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 = 2^{3} × 3^{4}

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 = 2^{3}

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 = 2^{3} × 3^{3}

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

Now the cube root is:

2^{1} × 3^{1} = 2 × 3 = 6

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