Subject: Compulsory Maths

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.

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 a^{3}= 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**

- Find the cube root of 2×2×2×3×3×3

= 2 × 3

= 6 - 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

- 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.

Find the cube of 6

Solution:

cube of 6

= 6^{3}

= 6×6×6

= 216

Find the cube of 16

Solution:

Cube of 16

= 16^{3}

=16 ×16 ×16

= 4096

Find the cube root of 125

Solution:

cube root of 125

=\(\sqrt[3]{125}\)

=\(\sqrt[3]{5×5×5}\)

=\(\sqrt[3]{5^3}\)

= 5

5 | 125 |

5 | 25 |

5 |

Find the cube of 20

Solution:

Cube of 20

= 20^{3}

= 20 × 20 × 20

= 8000

Find the cube of the following number:

35

Soln: Cube of 35=(35)^{3}

=35×35×35

=42875

Find the cube of 400

Solution:

Cube of 400

= 400^{3}

= 400×400×400

= 64000000

Find the smallest number by which the numbes must be divided to obtain a perfect square.

81

Solution:

3 | 81 |

3 | 27 |

3 | 9 |

3 |

81 = 3×3×3 =3^{3}

∴ The required number is 3.

Find the smallest number by which the number must be divided to obtain a perfect square.

182

Solution:

2 | 128 |

2 | 64 |

2 | 32 |

2 | 16 |

2 | 8 |

2 | 4 |

2 |

128 = 2^{3}×2^{3}×2

∴The required number is 2.

Find the smallest number by which the number must be divided to obtain a perfect square.

135

Solution:

3 | 135 |

3 | 45 |

3 | 15 |

5 |

135 = 3^{3}× 5

∴The required number is 5.

Find the cube root of 1331

Solution:

Cube root of 1331

= \(\sqrt{1331}\)

= \(\sqrt[3]{11 ×11 ×11}\)

= 11

11 | 1331 |

11 | 121 |

11 |

Find the smallest number by which the number must be multiplied to obtain a perfect cube.

243

Solution:

3 | 243 |

3 | 81 |

3 | 27 |

3 | 9 |

3 |

243 = 3^{3}× 3^{2}

∴The required number is 3.

Find the smallest number by which the number must be divided to obtain a perfect square.

192

Solution:

2 | 192 |

2 | 96 |

2 | 48 |

2 | 24 |

2 | 12 |

2 | 6 |

3 |

192 = 2^{3}×2^{3}×3

∴The required number is 3.

Find the smallest number by which the number must be divided to obtain a perfect square.

625

Solution:

5 | 625 |

5 | 125 |

5 | 25 |

5 |

325 = 5^{3}×5

∴The required number is 5.

Find the smallest number by which the number must be multiplied to obtain a perfect cube.

675

Solution:

3 | 675 |

3 | 225 |

3 | 75 |

5 | 25 |

5 |

675 = 3^{3}×5^{2}

∴The required number is 5.

Find the cube root of 4096.

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

Find the cube root of 2744.

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.

Find the cube root of 3375.

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.

Find the cube root of 10648.

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.

Simplify: \(\sqrt[3]{216}\) ÷ \(\sqrt[3]{64}\)

Solution:

2 | 216 |

2 | 108 |

2 | 54 |

3 | 27 |

3 | 9 |

3 |

2 | 64 |

2 | 32 |

2 | 16 |

2 | 8 |

2 | 4 |

2 |

\(\sqrt[3]{\frac{2×2×2×3×3×3}{2×2×2×2×2×2}}\)

= \(\sqrt[3]{\frac{3^3}{2^3}}\)

= \(\frac{3}{2}\)

Simplify: \(\sqrt[3]{8}\) + \(\sqrt[3]{27}\)

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

Simplify:

\(\sqrt[3]{27}\) × \(\sqrt[3]{-27}\).

Solution:

\(\sqrt[3]{3×3×3}\)× \(\sqrt[3]{-3×-3×-3}\)

= \(\sqrt[3]{3^3}\)× \(\sqrt[3]{(-3^3}\))

= 3×(-3)

= -9

Simplify: \(\sqrt[3]{8}\) + \(\frac{1}{\sqrt[3]{8}}\)

Solution:

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

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

= 2 + \(\frac{1}{2}\)

= \(\frac{2×2 +1×2}{2}\)

= \(\frac{5}{2}\)

Simplify: \(\sqrt[3]{8}\) - \(\frac{1}{\sqrt[3]{27}}\).

Solution:

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

= 3 - \(\frac{1}{3}\)

= \(\frac{3×3-1×1}{3}\)

= \(\frac{9-1}{3}\)

= \(\frac{8}{3}\)

Find the cube root of 400.

Solution:

Cube root of 400

= (400)^{3}

= 400 × 400 × 400

= 64,000,000

∴ Cube root of 400 is 64,000,000.

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