Cubes and Cube Roots

Here we will learn about special numbers, Cube and Cube roots. We obtain a cube number by multiplying the number with itself 3 times. Finding Cube Roots are inverse operation of cubing.

Cube Roots
Image Credit: Pixabay.com

Cube Number

➤ Product we get when a number is multiplied three times by itself is called a cube number.
or result we obtain when a number is cubed (raised to power of 3) is called a cube number.

2\times 2\times 2 = 8 ,

\frac{4}{3} \times \frac {4}{3} \times \frac {4}{3} = \frac {64}{27}\\  \ 3^3 = 27,\\  \ (-5)^3 = - 125\, etc.

Perfect Cube

When we multiply an integer three times by itself, cube number we get is called perfect cube.

So, a perfect cube is a cube of an integer.

(−2)×(−2)×(−2) = −8,

(−1)×(−1)×(−1) = −1,

1×1×1 = 1,

2×2×2 = 8,

3×3×3 = 27, etc
Cube of a negative integer is always a negative integer.
Cube of a positive integer is always a positive integer.

➤ Here, we will consider Cubes of only positive integers i.e, natural numbers,

1×1×1 = 1,

2×2×2 = 8,

3×3×3 = 27, etc

Properties of Cube numbers

Number Cube Number Cube
1

2

3

4

5

6

7

8

9

10

1

8

27

64

125

216

343

512

729

1000

11

12

13

14

15

16

17

18

19

20

1331

1728

2197

2744

3375

4096

4913

5832

6859

8000

→ The cube of an even number is always an even number and the cube of a odd number is always a odd number.

8^3 = 512, 16^3 = 4096\\  \ 7^3 = 343, 15^3 = 3375

Comparing Unit digits of a Number and its Cube

→ Cube of a number ending in digit 1, will end in digit 1.

1^3 = 1, 11^3 = 1331

→ Cube of a number ending in digit 2, will end in digit 8.

2^3 = 8, 12^3 = 1728

→ Cube of a number ending in digit 3, will end in 7.

3^3 = 27, 13^3 = 2197

→ Cube of a number ending in digit 4, will end in digit 4.

4^3 = 64, 14^3 = 2744

→ Cube of a number ending in digit 5, will end in digit 5.

5^3=125, 15^3 = 3375

→ Cube of a number ending in digit 6, will end in digit 6.

6^3 = 216, 16^3 = 4096

→ Cube of a number ending in digit 7, will end in digit 3.

7^3 = 343, 17^3 = 4913

→ Cube of a number ending in digit 8, will end in digit 2.

8^3 = 512, 18^3 = 5832

→ Cube of a number ending in digit 9, will end in digit 9.

9^3 = 729, 19^3 = 6859

→ Cube of a number ending in digit 0, will end in digit 0.

10^3 = 1000, 20^3 = 8000

☆☆ From above we can observe that the Cube of  the numbers ending in digits 0, 1, 4, 5, 6 and 9 are the numbers ending in same digits.

→ Cube numbers can only have odd number of zeros at the end which must be triple the number of zeros at the end of the number whose cube it is.

10^3 =1000, 20^3 = 8000, 30^3 =27000, 1000^3 = 1000000000

Cube Roots

↪ Finding cube roots are the inverse operation of cubing.

Cube root of a perfect cube number is the number whose perfect cube the number is.

8 = 2×2×2, so we say that cube root of 8 is 2.

27 = 3×3×3, so we say that cube root of 27 is 3.

→ The symbol \sqrt[3] denotes the cube root.

\sqrt[3]{8} = 2,\, \sqrt[3]{27}= 3

Statement Inference Statement Inference
1^3 =1 \sqrt[3]1 = 1 6^3 =216 \sqrt[3]{216} = 6
2^3 =8 \sqrt[3]8 = 2 7^3 =343 \sqrt[3]343 = 7
3^3 =27 \sqrt[3]27 = 3 8^3 =512 \sqrt[3]512 = 8
4^3 =64 \sqrt[3]64 = 4 9^3 =729 \sqrt[3]729 = 9
5^3 =125 \sqrt[3]125 = 5 10^3 =1000 \sqrt[3]1000 = 10

Finding Cube roots

Prime Factorisation method

→ Write the given number as product of its prime factors

→ Group the prime factors in triples.

→ Express the tripled prime factors in exponential form of power of 3

→ Take the power of 3 as common

→ Put \sqrt[3] before the number in LHS and remove the power in RHS to get the required value.

e.g. 216 = 2×2×2×3×3×3

⇒ 216 = 2×2×2×3×3×3

\implies 216 = 2^3 \times 3^3\\  \implies 216 = (2 \times 3)^3\\  \implies \sqrt[3]{216} = 2\times 3 = 6\\  \therefore \sqrt[3]{216} = 6

Problems with Solution

• Problems on Cube Numbers
– Finding Cube of a Number
– Finding the least Number to multiply / divide from a number to obtain a perfect cube.
– Word Problems.

• Problems on Cube Roots
– Finding Cube Roots by Factorisation Method
– Estimating Cube Roots by grouping digits of Numbers.

Leave a Reply

Close Menu
%d bloggers like this: