What is Cubes and Cube Roots Class 8: Definition & Concepts

Understanding Cubes: Definition and Properties

A cube of a number $n$ is the product of $n$ multiplied by itself three times. Mathematically, it is expressed as:

$$ \text{Cube of } n = n^3 = n \times n \times n $$

For example, the cube of 4 is:

$$ 4^3 = 4 \times 4 \times 4 = 64 $$

Properties of Cubes:

• Cubing a positive number gives a positive result.
• Cubing a negative number gives a negative result.
• Cubes grow faster than squares as numbers increase.
• The units digit of cubes follows a specific pattern depending on the units digit of the base number.

Knowing cubes helps in solving volume problems and algebraic expressions in Class 8 NCERT Maths.

What is Cube Root? Definition and How to Find It

The cube root of a number $x$ is a number $y$ such that when $y$ is cubed, it equals $x$:

$$ \sqrt[3]{x} = y \quad \text{if} \quad y^3 = x $$

For example:

$$ \sqrt[3]{125} = 5 \quad \text{because} \quad 5^3 = 125 $$

Methods to find cube roots:

• Prime Factorisation: Break the number into prime factors and group them in triples.
• Estimation: Use nearby perfect cubes to estimate.
• Calculator: Use the cube root function for large numbers.

Cube roots are important for reversing cube operations and solving equations.

Perfect Cubes and Their Cube Roots: A Comparison Table

Memorising perfect cubes and their cube roots helps solve problems quickly. Here is a comparison table of cubes from 1 to 12:

Use this table to quickly identify cubes and cube roots during exams.

How to Calculate Cube Roots Using Prime Factorisation

Prime factorisation is a reliable method to find the cube root of a perfect cube.

Steps:

1. Express the number as a product of prime factors. 2. Group the prime factors into triples. 3. Take one factor from each group to form the cube root.

Example: Find the cube root of 216.

• Prime factorisation:

$$216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 = 2^3 \times 3^3$$

• Grouping into triples: $(2^3)(3^3)$
• Taking one from each group: $2 \times 3 = 6$

So,

$$ \sqrt[3]{216} = 6 $$

This method works well for perfect cubes and helps in understanding the structure of numbers.

Worked Examples on Cubes and Cube Roots

Example 1: Find the cube of 7.

Solution:

$$ 7^3 = 7 \times 7 \times 7 = 343 $$

Example 2: Find the cube root of 512.

Solution:

Using prime factorisation:

$$ 512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^9 $$

Group into triples:

$$ (2^3)(2^3)(2^3) = (2 \times 2 \times 2)^3 $$

Cube root:

$$ \sqrt[3]{512} = 2^3 = 8 $$

These examples show how to apply formulas and methods to solve problems easily.

Applications of Cubes and Cube Roots in Class 8 Maths

Understanding cubes and cube roots is essential in various areas of Class 8 Mathematics:

• Geometry: Calculating volumes of cubes and cuboids.
• Algebra: Simplifying expressions involving powers.
• Number Theory: Recognising perfect cubes and their properties.
• Real-life Problems: Estimating quantities in three dimensions.

For example, the volume $V$ of a cube with side length $a$ is:

$$ V = a^3 $$

Knowing cube roots helps find the side length if volume is given:

$$ a = \sqrt[3]{V} $$

Mastering these concepts strengthens your problem-solving skills for exams.