Cubes and Cube Roots
What is a perfect cube?
What is Cube Root?
0 cubed

=

0^{3}

=

0 × 0 × 0

=

0

1 cubed

=

1^{3}

=

1 × 1 × 1

=

1

2 cubed

=

2^{3}

=

2 × 2 × 2

=

8

3 cubed

=

3^{3}

=

3 × 3 × 3

=

27

4 cubed

=

4^{3}

=

4 × 4 × 4

=

64

5 cubed

=

5^{3}

=

5 × 5 × 5

=

125

6 cubed

=

6^{3}

=

6 × 6 × 6

=

216

Perfect Cubes
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

Finding the Cube Root of a Number by Prime Factorisation
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
singledigit number can have a maximum of 3 digits only. (Remember, 93 = 729)
Therefore, the
cube root of any number less than 1000 is a singledigit number.
The cube of a twodigit 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 twodigit 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 fourdigit number, fivedigit number or a sixdigit
number can be a twodigit 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., 2^{3} < 19 < 3^{3}
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 7^{3}
= 343 and 8^{3} = 512
∴ 343 < 474
< 512, i.e., 7^{3} < 474 < 8^{3}
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.