## 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.

**Example:**

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.

**Example:**

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.

## 4. 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.

## 5. 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*