## Cube Root

Subject: Compulsory Maths

#### Overview

To cube a number, just use it in a multiplication 3 times.A cube root is a number that multiplies by itself three times in order to create a cubic value.In some contexts, particularly when the number, one of the cube roots ( in this particular case the real one) is referred to as the principal cube root.

##### Cube Root

To find cube root, make triple of equal factors. The opposite of cubing a number is called finding the cube root. A cube root is a number, that is multiplied by itself three times in order to create a cubic value. A cube root of a number x is a number, such that a3= x. All real numbers (except zero) have exactly one real cube root.

Cube of 6 = 6³ =216

Cube root of 216 = 6

Examples

• The cube root of 64 is 4 ( because 4x4x4=64)
• The cube root of 125 is 5 ( because 5x5x5=125)
• The cube root of 512 is 8 ( because 8x8x8=512 )

The symbol, $\sqrt [3]{}$, means cube root, so $\sqrt [3]{27}$ means "cube root of 27" and $\sqrt[3]{64}$means "Cube root of 64"

Thus $\sqrt [3]{27}$ = $\sqrt [3]{3^3}$ = 3 and $\sqrt[3]{64}$ = $\sqrt[3]{4^3}$ = 4

A natural number is known as a perfect cube or a cube number.

Cube root of a perfect cube can be found by factorization method.

• The number should be the factor of the prime number or should be expressed as the factor of the prime number.
• Make triples of the factor and each triple should be equal.
• Take one factor from each triple.
• The product is the cube root of the given number.

Examples

1. Find the cube root of 2×2×2×3×3×3
= 2 × 3
= 6

2. Find the cube root of 729.
Solution:
$\sqrt[3]{729}$
= $\sqrt[3]{3×3×3×3×3×3}$
= $\sqrt[3]{3^3×3^3}$
= 3×3
= 9
##### Things to remember
• A cube root is a number, that multiplied by itself three times in order to create a cubic value.
• To find cube root, make triple of equal factors.
• The opposite of cubing a number is called finding the cube root.
• It includes every relationship which established among the people.
• There can be more than one community in a society. Community smaller than society.
• It is a network of social relationships which cannot see or touched.
• common interests and common objectives are not necessary for society.

Solution:

cube of 6

= 63

= 6×6×6

= 216

Solution:

Cube of 16

= 163

=16 ×16 ×16

= 4096

Solution:

cube root of 125

=$\sqrt[3]{125}$

=$\sqrt[3]{5×5×5}$

=$\sqrt[3]{5^3}$

= 5

 5 125 5 25 5

Solution:

Cube of 20

= 203

= 20 × 20 × 20

= 8000

Solution:

Cube of 400

= 4003

= 400×400×400

= 64000000

Solution:

 3 81 3 27 3 9 3

81 = 3×3×3 =33

∴ The required number is 3.

Solution:

 2 128 2 64 2 32 2 16 2 8 2 4 2

128 = 23×23×2

∴The required number is 2.

Solution:

 3 135 3 45 3 15 5

135 = 33× 5

∴The required number is 5.

Solution:

Cube root of 1331

= $\sqrt{1331}$

= $\sqrt[3]{11 ×11 ×11}$

= 11

 11 1331 11 121 11

Solution:

 3 243 3 81 3 27 3 9 3

243 = 33× 32

∴The required number is 3.

Solution:

 2 192 2 96 2 48 2 24 2 12 2 6 3

192 = 23×23×3

∴The required number is 3.

Solution:

 5 625 5 125 5 25 5

325 = 53×5

∴The required number is 5.

Solution:

 3 675 3 225 3 75 5 25 5

675 = 33×52

∴The required number is 5.

Solution:

 2 4096 2 2048 2 1024 2 512 2 256 2 128 2 64 2 32 2 16 2 8 2 4 2

= $\sqrt{2×2×2×2×2×2×2×2×2×2×2×2}$

= $\sqrt[3]{2^3×2^3×2^3×2^3}$

=2 × 2 × 2 × 2

= 16

Solution:

 2 2744 2 1372 2 686 7 343 7 49 7

= $\sqrt[3]{2×2×2×7×7×7}$

= $\sqrt[3]{2^3×7^3}$

= 2 × 7

= 14

∴ The Cube root of 2744 is 14.

Solution:

 3 3375 3 1125 3 375 5 125 5 25 5

= $\sqrt[3]{3×3×3×5×5×5}$

= $\sqrt[3]{3^3×5^3}$

= 3 × 5

= 15

∴ The Cube root of 3375 is 15.

Solution:

 2 10648 2 5324 2 2662 11 331 11 121 11

= $\sqrt[3]{2×2×2×11×11×11}$

= $\sqrt[3]{2^3×11^3}$

= 2 × 11

= 22

∴ Cube root of 10648 is 22.

Solution:

 2 216 2 108 2 54 3 27 3 9 3

 2 64 2 32 2 16 2 8 2 4 2

$\sqrt[3]{\frac{2×2×2×3×3×3}{2×2×2×2×2×2}}$

= $\sqrt[3]{\frac{3^3}{2^3}}$

= $\frac{3}{2}$

Solution:

 2 8 2 4 2

 3 27 3 9 3

= $\sqrt[3]{2×2×2}$ + $\sqrt[3]{3×3×3}$

=$\sqrt[3]{2^3}$ + $\sqrt[3]{3^3}$

= 2 + 3

= 5

Solution:

$\sqrt[3]{3×3×3}$× $\sqrt[3]{-3×-3×-3}$

= $\sqrt[3]{3^3}$× $\sqrt[3]{(-3^3}$)

= 3×(-3)

= -9

Solution:

$\sqrt[3]{2×2×2}$ + $\frac{1}{\sqrt[3]{2×2×2}}$

= $\sqrt[3]{2^3}$ + $\frac{1}{\sqrt[3]{2^3}}$

= 2 + $\frac{1}{2}$

= $\frac{2×2 +1×2}{2}$

= $\frac{5}{2}$

Solution:

= $\sqrt[3]{3×3×3}$ - $\frac{1}{\sqrt[3]{3×3×3}}$

= 3 - $\frac{1}{3}$

= $\frac{3×3-1×1}{3}$

= $\frac{9-1}{3}$

= $\frac{8}{3}$

Solution:

Cube root of 400

= (400)3

= 400 × 400 × 400

= 64,000,000

∴ Cube root of 400 is 64,000,000.