Squares and Square Roots Class 8 Solutions: Clear Concepts & Examples

Understanding Squares and Square Roots

In Class 8 Mathematics, the chapter on squares and square roots introduces two fundamental concepts:

• Square of a number: The square of a number $n$ is $n \times n$, written as $n^2$. For example, the square of 5 is $5^2 = 25$.
• Square root of a number: The square root of a number $x$ is a value $y$ such that $y^2 = x$. For example, the square root of 25 is 5 because $5^2 = 25$.

These concepts form the basis for many mathematical operations and problem-solving techniques. Understanding them helps in simplifying expressions and solving equations.

Key Definitions

• Perfect Square: A number that is the square of an integer (e.g., 1, 4, 9, 16).
• Non-Perfect Square: Numbers that do not have an integer as their square root (e.g., 2, 3, 5).

This chapter also covers methods to find squares and square roots efficiently, which are important for exams and real-life applications.

Methods to Calculate Squares and Square Roots

Class 8 NCERT solutions teach you several methods to calculate squares and square roots:

Calculating Squares

• Direct multiplication: Multiply the number by itself.
• Using algebraic identities: For example, $(a + b)^2 = a^2 + 2ab + b^2$.

Calculating Square Roots

• Prime Factorization Method:

1. Factorize the number into prime factors. 2. Pair the prime factors. 3. Take one factor from each pair. 4. Multiply these factors to get the square root.

• Long Division Method: A step-by-step division process to find square roots of large numbers or decimals.

#### Worked Example: Prime Factorization Find the square root of 144.

Prime factors of 144 = $2 \times 2 \times 2 \times 2 \times 3 \times 3$

Pairing factors: $(2 \times 2), (2 \times 2), (3 \times 3)$

Taking one from each pair: $2 \times 2 \times 3 = 12$

So, $\sqrt{144} = 12$.

Perfect Squares and Their Properties

Perfect squares are numbers like 1, 4, 9, 16, 25, and so on, which are squares of whole numbers. Knowing perfect squares up to 20 helps you solve problems faster.

Properties of Perfect Squares

• They end with digits 0, 1, 4, 5, 6, or 9.
• The square root of a perfect square is always an integer.
• Sum of the first $n$ odd numbers is $n^2$.

Memorizing these squares helps in quick calculations during exams.

Using Squares and Square Roots in Problem Solving

Squares and square roots are used in various mathematical problems, including:

• Simplifying algebraic expressions
• Solving quadratic equations
• Calculating areas and volumes
• Finding distances using the Pythagoras theorem

Example Problem

Find the length of the diagonal of a square whose side is 7 cm.

Solution:

Diagonal $d = \sqrt{side^2 + side^2} = \sqrt{7^2 + 7^2} = \sqrt{49 + 49} = \sqrt{98}$

Simplify $\sqrt{98}$:

$98 = 49 \times 2$, so $\sqrt{98} = \sqrt{49} \times \sqrt{2} = 7\sqrt{2}$ cm.

Thus, the diagonal is $7\sqrt{2}$ cm.

This example shows how squares and square roots help solve geometry problems.

Tips to Master Squares and Square Roots for Class 8 Exams

To excel in the squares and square roots chapter, follow these tips:

• Understand concepts: Don’t just memorize formulas; understand why they work.
• Practice regularly: Solve all NCERT exercises and additional problems.
• Memorize squares: Know squares of numbers up to 20 by heart.
• Use shortcuts: Learn algebraic identities to calculate squares quickly.
• Check your work: Always verify your answers to avoid careless mistakes.

Consistent practice with NCERT solutions builds confidence and improves problem-solving speed, essential for CBSE Class 8 exams.