Cubes and Cube Roots

What is a perfect cube?

A perfect cube is a number that is a cube of an integer.

For example, 125 is a perfect cube since 125 = 5 × 5 × 5 = 5³.

Some examples of perfect cubes are 1, 8, 27, 64, 125, 216, 343.

What is Cube Root?

Finding the cube root is opposite the cubing of a number. Since 5³ = 125, cube root of 125 is 5. Cube root of a perfect cube is an integer.

Again, 3 cubed is 27, so the cube root of 27 is 3.

It is possible to obtain the cube root of a negative number. For example, the cube root of −125 is −5.

Cubes From 0³ to 6³

0 cubed	=	0³	=	0 × 0 × 0	=	0
1 cubed	=	1³	=	1 × 1 × 1	=	1
2 cubed	=	2³	=	2 × 2 × 2	=	8
3 cubed	=	3³	=	3 × 3 × 3	=	27
4 cubed	=	4³	=	4 × 4 × 4	=	64
5 cubed	=	5³	=	5 × 5 × 5	=	125
6 cubed	=	6³	=	6 × 6 × 6	=	216

Perfect Cubes

The Perfect Cubes are the cubes of the whole number:

Number	Perfect Cubes
0	0
1	1
2	8
3	27
4	64
5	125
6	216
7	343
8	512
9	729
10	1000
11	1331
12	1728
13	2197
14	2744
15	3375

It is easy to work out the cube root of a perfect cube, but it is really hard to work out other cube roots.

Finding the Cube Root of a Number by Prime Factorisation

Example 1: Find the cube root of 2744.

Solution: We write 2744 as the product of its prime factors.

2744 = 2 × 2 × 2 × 7 × 7 × 7

Thus, the cube root of 2744 is 2 × 7, that is, 14.

Example 2: Find the cube root of 42,875.

Solution: We write 2744 as the product of its prime factors.

42,875 = 5 × 5 × 5 × 7 × 7 × 7

Thus, the cube root of 42,875 is 5 × 7, that is, 35.

Example 3: Find the cube root of 13824.

Solution: We write 13824 as the product of its prime factors.

13824 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3

Thus, the cube root of 13824 is 2 × 2 × 2 × 3, that is, 24.

Finding the Cube Root of a Number by Estimation

The cube of a single-digit number can have a maximum of 3 digits only. (Remember, 93 = 729)

Therefore, the cube root of any number less than 1000 is a single-digit number.

The cube of a two-digit number can have a maximum of 6 digits. (Because 100 × 100 × 100 = 1000000, which is the smallest 7- digit number.)

This means the cube of a two-digit number can have 4 digits, 5 digits and 6 digits at the most.

99 × 99 × 99 = 970299 (6 digits)

25 × 25 × 25 = 15625 (5 digits)

10 × 10 × 10 = 1000 (4 digits)

It follows that the cube root of a four-digit number, five-digit number or a six-digit number can be a two-digit number only.

Example 1: Find the cube root of 19683 by estimation.

Solution: Form groups of three starting from the rightmost digit of 19683.

19 683

Second group First group

19 683

Consider the first group 683.

The ones digit is 3. So, the required cube root of the given number ends with 7 in its ones place.

Now, take the next group 19.

The cube of 2 is 8 and cube of 3 is 27.

∴ 19 lies between 8 and 27, i.e., 2³ < 19 < 3³

We take the smaller number between 2 and 3 for the tens place of the required number.

Thus, we take 2 in the tens place of the required number.

Hence, the cube root of 19683 is 27.

Example 2: Find the cube root of 474552 using estimation.

Solution: Make groups of three digits starting from the right hand side.

474 552

Second group First group

474 552

First group is 552.

The ones digit is 2. Hence, the cube root of the given number ends with 8 in its ones place. Now, consider the next group 474.

We know that 7³ = 343 and 8³ = 512

∴ 343 < 474 < 512, i.e., 7³ < 474 < 8³

The smaller number between 7 and 8 is 7.

We take this 7 in the tens place of the required number.

Hence, the cube root of 474552 is 78.

Cubes and Cube Roots