# Class-8 Cubes And Cube Roots

Finding Cube Roots by Prime Factorization

Cubes And Cube Roots Worksheets

## Introduction

In previous chapter, we learnt about square and square root, now we use the knowledge that we gain from square and square root to cube and cube roots in this chapter. In previous chapter we found out square root of positive number, but cube roots are found for positive as well as negative numbers.

## Cubes

When same three natural numbers multiplied itself three times to get cube number. Cube of a natural number is the number raised to the power 3.

So, 6^{3}= 6 × 6 × 6

Number | Cube |
---|---|

1^{3} |
1 |

2^{3} |
8 |

3^{3} |
27 |

4^{3} |
64 |

5^{3} |
125 |

6^{3} |
216 |

7^{3} |
343 |

8^{3} |
512 |

9^{3} |
729 |

10^{3} |
1000 |

## Perfect Number

If a natural number is P can be expressed as R^{3}, here R is a natural number, then P is a cube number or perfect cube.
Hence, cubes of natural numbers are called perfect cube.

**Example.** 2^{3} = 8, 7^{3} = 343, 9^{3} = 729.

But all-natural numbers are not perfect cubes. A perfect cube can always made by the product of triplets of equal prime number. Some examples are provided below to understand how to figure out a perfect cube numbers and a non-perfect cube numbers.

**Example 1.** Check if 512 is a perfect cube number.

**Solution.** First express 512 into prime factors, 512 = 2×2×2×4×4×4

Then group each prime numbers into triplet as shown below.

__2×2×2__×__4×4×4__

so, here 512 is expressed as multiplication of triplets equal prime number.

**Example 2.** Check if 135 ia a perfect cube number.

**Solution.** 135 = 3×3×3×5×5

Here 135 can not be expressed in all prime numbers triplets form. One more 5 is required to form triplets.

So, 135 is not a cube number.

To make perfect cube, we should multiply or divide the left unpaired prime number with non-cube number. In the above example, we need one more 5 to make triplets of 5. We can conclude that, if we muliiply 5 with 135, then it will become a perfect cube.

## Properties of Cubes

- Cubes of all odd numbers are odd
- Cubes of all even numbers are even
- Unit's place number rule

### Cubes of all odd numbers are odd

3^{3} = 27, 5^{3} = 125, 7^{3} = 343, 9^{3} = 729, (729)^{3} = 38,74,20,489

So, we can conclude that cubes of an odd number is odd.

### Cubes of all even numbers are even

8^{3} = 512, 32^{3} = 32786, 10^{3} = 1000, 18^{3} = 5832, 22^{3} = 10,648

So, We can conclude that cubes of an even number is even.

### Unit's place number rule

If a number has 2 in the unit's place, then its cube ends with 8.

If a number has 3 in the unit's place, then its cube ends with 7.

If a number has 8 in the unit's place, then its cube ends with 2.

If a number has 7 in the unit's place, then its cube ends with 3.

If a number has 0 in the unit's place, then its cube ends with three zeros.

If a number has 1, 4, 5, 6 or 9 in the unit place, then its cube ends same digits.

## Cubes of Negative Integer

Cubes of negative integers always formed negative integers. Let's see some examples.

**Example 1.** Given number is (-3)^{3}.

**Solution.** (−3)^{3} = (−3) × (−3) × (−3) = −27

**Example 2.** Given number is (−5)^{3}.

**Solution.** (−5)^{3} = (−5) × (−5) × (−5) = −125

## Cubes of Rational Number

If ^{a}⁄_{b} is a rational number and b ≠ 0 ,then (^{a}⁄_{b})3 = (^{a}⁄_{b}) × (^{a}⁄_{b}) × (^{a}⁄_{b}) = ^{a3}⁄_{b3}.
Hence, (^{a}⁄_{b})^{3} = ^{a3}⁄_{b3}.

**Example 1.** Find the cube of ^{2}⁄_{3}.

**Solution.** (^{2}⁄_{3})^{3} = ^{23}⁄_{33} = ^{(2x2x2)}⁄_{(3x3x3)} = ^{8}⁄_{27}

**Example 2.** Find the cube of ^{-2}⁄_{5}.

**Solution.** (^{-2}⁄_{5})^{3} = ^{-23}⁄_{53} = ^{{(-2)x(-2)x(-2)}}⁄_{(5x5x5)} = ^{-8}⁄_{125}

## Cube Roots

The cube root of a number (R) is a number (S) that multiplied with same number three times (S) get again the number (R).

If S is the cube root of number R ,then S × S × S = R or S^{3} = R.

The cube root of number R is represented by ∛ R .

Hence, S = ∛ R .

**Notes:** Calculating cube root is inverse operation of calculating the cube.

4^{3} = 64
=> ∛ 64 = 4

## Finding Cube Roots using Prime Factorization

- Finding cube roots of natural number
- Finding cube roots of negative perfect cube
- Finding cube root of product of integers
- Finding cube root of a rational number
- Finding cube root of decimal number

### Finding cube roots of natural number

Step 1. The given number should be expressed in the form of product of prime factors

Step 2. Make the prime numbers in groups in triplets of same prime number

Step 3. Consider one prime number from each triplet of primes and multiply them together. The product is equal to the required number of cube root of the given number.

**Example 1.** Find the cube root of 5832 by prime factorisation.

**Solution.** Given number is 5832.

Express 5832 into prime factorisation form.

5832 = __2×2×2__×__3×3×3__×__3×3×3__

∛ 5832 = 2 × 3 × 3 = 18

Hence, cube root of 5832 is 18.

### Finding cube roots of negative perfect cube

As we know (−P)^{3} = (−P) × (−P) × (−P) = (−P)^{3}

Then ∛(−P)^{3} = −∛(−P)^{3} = −P.

So, for better understanding we should write that, cube root of (−a^{3}) = −(cube root of a^{3})

**Example 1.** Find the cube root of (−729)

**Solution.** Given number is −729.

As we know, ∛ -729 = -∛ -729

First we express 729 into prime factors by prime factorisation method, we get

729 = __3×3×3__×__3×3×3__

∛ 729 = 3 × 3 = 9

Hence, ∛ -729 = -9

### Finding cube root of product of integers

If we take two integers like p and q, then

∛ pq = (pq)^{1⁄3} = (p)^{1⁄3} × (q)^{1⁄3}

= ∛ p × ∛ q

Let's see one example of product of two integers below to better understand.

**Example 1.** Calculate ∛ {512 × (-64)} .

**Solution.** By using formula ∛ pq = ∛ p × ∛ q

∛ {512 × (-64)}

= ∛ 512 × ∛ -64

= ∛ (8 × 8 × 8) × {-∛ (4 × 4 × 4) }

= 8 × (-4)

= -32

Hence, cube root of product of 512 and -64 is -32.

### Finding cube root of a rational number

If we take a cube root of a rational number ^{m}⁄_{n}, then

∛ m/n = (^{m}⁄_{n})^{1⁄3} = ^{∛ m }⁄_{∛ n }

let's take one example to understand this.

**Example 1.** Calculate ∛ 8/27 .

**Solution.** ∛ 8/27

= ^{∛ 8 }⁄_{∛ 27 }

= ^{∛ 2x2x2 }⁄_{∛ 3x3x3 }

= ^{2}⁄_{3}

Hence, cube root of ^{8}⁄_{27} is ^{2}⁄_{3}.

### Finding cube root of decimal number

To find out cube root of decimal number, first we should convert decimal number to fraction in it's lowest term. Let's see one example to understand it better.

**Example 1.** Find the cube roots of 19.683.

**Solution.** ∛ 19.683

= ∛ 19683/1000

= ^{∛ 19683 }⁄_{∛ 1000 }

= ^{∛ (3×3×3×3×3×3×3×3×3) }⁄_{∛ 10×10×10 }

= ^{(3×3×3)}⁄_{10}

= ^{27}⁄_{10}

= 2.7

Hence, cube root of 19.683 is 2.7.

## Class-8 Cubes And Cube Roots Test

## Class-8 Cubes And Cube Roots Worksheets

Cubes And Cube Roots Worksheet - 1

Cubes And Cube Roots Worksheet - 2

Cubes And Cube Roots Worksheet - 3

## Answer Sheet

**Cubes-And-Cube-Root-Answer**Download the pdf

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