Cubes and Cube Roots Chapter Notes | Mathematics Class 8 ICSE PDF Download

Introduction

This chapter introduces the concepts of cubes and cube roots, which are essential in understanding higher powers and their inverses in mathematics. A cube of a number is obtained by multiplying the number by itself three times, and the cube root is the reverse process, finding the number that, when cubed, gives the original value. These concepts help in solving problems involving volumes, factorisation, and number properties.

What is a Cube?

• A cube of a number is the result of multiplying that number by itself three times.
• It is denoted as m³, where m is the number.

Steps to find a cube:

• Take a number, say m.
• Multiply the number m by itself three times: m × m × m = m³.

Examples:

• Cube of 5 = 5 × 5 × 5 = 125, so 5³ = 125.
• Cube of 8 = 8 × 8 × 8 = 512, so 8³ = 512.
• Cube of -4 = -4 × -4 × -4 = -64, so (-4)³ = -64.

Key points:

• Cube of a positive number is always positive.
• Cube of a negative number is always negative.

Perfect Cube

• A number is called a perfect cube if it can be written as the product of an integer multiplied by itself twice more — that is, three times in total. In other words, a number n is a perfect cube if there exists an integer a such that n = a × a × a or n = a³. For example, 27 is a perfect cube because 3 × 3 × 3 = 27.
• On the other hand, a non-perfect cube is a number that cannot be written as the product of an integer multiplied by itself three times. This means there is no whole number whose cube equals that number. For example, 20 is a non-perfect cube because there is no integer a such that a × a × a = 20.

Steps to check if a number is a perfect cube

• Identify the prime factors of the number.
• Arrange the prime factors into groups of three identical factors.
• If all prime factors can be grouped into complete sets of three, the number is a perfect cube.

Example 1: For 216

• Prime factors: 216 = 2 × 2 × 2 × 3 × 3 × 3.
• Group as: (2 × 3) × (2 × 3) × (2 × 3) = 6 × 6 × 6 = 6³.
• Since 216 = 6³, it is a perfect cube.

Example 2: For 297

• Prime factors: 297 = 3 × 3 × 3 × 11.
• Group as: (3 × 3 × 3) × 11.
• Since 11 does not form a triplet, 297 is not a perfect cube.

Example 3: For 2744

• Prime factors: 2744 = 2 × 2 × 2 × 7 × 7 × 7.
• Group as: (2 × 7) × (2 × 7) × (2 × 7) = 14 × 14 × 14 = 14³.
• Since 2744 = 14³, it is a perfect cube.

Finding the smallest number to make a perfect cube

• Factorise the number into primes.
• Identify any prime factors that do not form a complete triplet.
• Multiply by the missing factors to complete the triplets.

Example: For 3087

• Prime factors: 3087 = 3 × 3 × 7 × 7 × 7.
• Need one more 3 to form 3 × 3 × 3.
• Multiply by 3: 3087 × 3 = 3 × 3 × 3 × 7 × 7 × 7 = 21³, a perfect cube.

Finding the smallest number to divide to make a perfect cube

• Factor the number into its prime components.
• Identify any prime factors that are lacking to form a complete group of three.
• Multiply by the missing factors to complete these groups.

Example: For 6750

• Prime factors: 6750 = 2 × 5 × 5 × 5 × 3 × 3 × 3.
• Extra factor: 2 (since it does not form a triplet).
• To achieve a perfect cube, divide by the extra factors needed to form complete triplets.
• Divide by 2: 6750 ÷ 2 = 5 × 5 × 5 × 3 × 3 × 3 = 15³, a perfect cube.

Properties

The cube of an odd number is always odd. For example:

• 1³ = 1
• 3³ = 27
• 5³ = 125

The cube of an even number is always even. For example:

• 2³ = 8
• 4³ = 64
• 6³ = 216

Cube Roots

A cube root of a number x is another number y which, when multiplied by itself three times, gives back x. In mathematical terms, if y³ = x, then y is called the cube root of x. This is represented as ∛x.

Examples:

• Cube root of 27 is 3, since 3³ = 27, so ∛27 = 3.
• Cube root of 343 is 7, since 7³ = 343, so ∛343 = 7.
• Cube root of 125 is 5, since 5³ = 125, so ∛125 = 5.
• Cube root of 512 is 8, since 8³ = 512, so ∛512 = 8.

Cube Root by Factorisation

Steps to find the cube root by factorisation

• Identify Prime Factors: Start by identifying the prime factors of the given number.
• Group Factors: Group the prime factors into sets of three identical factors.
• Select Factors: Select one factor from each group of three.
• Multiply Factors: Multiply these selected factors together to find the cube root.

Example 1: For 216

• Prime factors: 216 = 2 × 2 × 2 × 3 × 3 × 3.
• Group as: (2 × 2 × 2) × (3 × 3 × 3).
• Take one factor from each: 2 × 3 = 6.
• So, ∛216 = 6.

Example 2: For 1728

• Prime factors: 1728 = (2 × 2 × 2) × (2 × 2 × 2) × (3 × 3 × 3).
• Take one factor from each triplet: 2 × 2 × 3 = 12.
• So, ∛1728 = 12.

Finding the smallest number to make a perfect cube

• Factorise the number into primes.
• Identify missing factors needed to complete triplets.
• Multiply by those factors.

Example: For 210125

• Prime factors: 210125 = 5 × 5 × 5 × 41 × 41.
• Need one more 41 to form 41 × 41 × 41.
• Multiply by 41: 210125 × 41 = (5 × 5 × 5) × (41 × 41 × 41) = 205³.

1. Cube-Root of a Negative Perfect Cube

• The cube root of a negative perfect cube is negative.
• If m is a positive integer, then (-m)³ = -m³.
• Thus, ∛(-m³) = -∛m³ = -m.

Examples:

• Cube root of -8 = -∛8 = -2, since (-2)³ = -8.
• Cube root of -1000 = -∛1000 = -10, since (-10)³ = -1000.
• Cube root of -1 = -∛1 = -1, since (-1)³ = -1.

2. Cube-Root of Product of Numbers

The cube root of a product equals the product of the cube roots.

Formulas:

• ∛(x × y) = ∛x × ∛y.
• ∛(x × y × z) = ∛x × ∛y × ∛z.

Examples:

• ∛(8 × 125) = ∛8 × ∛125 = 2 × 5 = 10.
• ∛(500 × 54) = ∛(1000 × 27) = ∛1000 × ∛27 = 10 × 3 = 30.
• ∛(-16 × 32) = ∛(-8 × 64) = ∛(-8) × ∛64 = -2 × 4 = -8.
• ∛(216 × -343) = ∛216 × ∛(-343) = 6 × -7 = -42.

3. Cube-Root of Fractional Numbers

The cube root of a fraction is the cube root of the numerator divided by the cube root of the denominator.

Formulas:

• ∛(x/y) = ∛x / ∛y.
• ∛(xy/z) = (∛x × ∛y) / ∛z.
• ∛(x/(y × z)) = ∛x / (∛y × ∛z).

Examples:

• ∛(125/216) = ∛125 / ∛216 = 5 / 6.
• ∛(-8/27) = ∛(-8) / ∛27 = -2 / 3.
• ∛(64/-125) = ∛64 / ∛(-125) = 4 / -5 = -4/5.

4. Cube Root of a Decimal Number

Steps to find the cube root of a decimal:

• Convert the decimal to a fraction.
• Find the cube root of the fraction using ∛(x/y) = ∛x / ∛y.
• Simplify the result.

Examples:

• ∛0.027 = ∛(27/1000) = ∛27 / ∛1000 = 3 / 10 = 0.3.
• ∛0.008 = ∛(8/1000) = ∛8 / ∛1000 = 2 / 10 = 0.2.
• ∛0.125 = ∛(125/1000) = ∛125 / ∛1000 = 5 / 10 = 0.5.