What is Squares and Square Roots Class 8: Definition & Concepts

Understanding Squares: Definition and Properties

In Class 8 Mathematics, a square of a number means multiplying the number by itself. For any number $a$, its square is written as $a^2$ and calculated as:

$$ a^2 = a \times a $$

Example:

• The square of 5 is $5^2 = 5 \times 5 = 25$
• The square of 12 is $12^2 = 144$

Properties of squares:

• Squares of positive and negative numbers are always positive.
• The square of zero is zero.
• Squares grow rapidly as numbers increase.

Squares are used to calculate areas of squares and in many algebraic expressions in Class 8.

Square Roots: Meaning and How to Find Them

A square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the radical symbol $\sqrt{}$.

If $b^2 = a$, then $b$ is the square root of $a$, written as:

$$ b = \sqrt{a} $$

Example:

• $\sqrt{25} = 5$ because $5^2 = 25$
• $\sqrt{144} = 12$

Square roots can be:

• Perfect square roots: When the number is a perfect square (like 25, 36).
• Non-perfect square roots: When the number is not a perfect square (like 20, 50), these roots are irrational and usually approximated.

Methods to find square roots include prime factorization and long division method.

Prime Factorization Method to Find Square Roots

The prime factorization method is a simple way to find the square root of a perfect square.

Steps: 1. Express the number as a product of prime factors. 2. Pair the prime factors. 3. Take one factor from each pair. 4. Multiply these factors to get the square root.

Example: Find $\sqrt{144}$

• Prime factors of 144:

$$144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3$$

• Group pairs:

$$(2 \times 2), (2 \times 2), (3 \times 3)$$

• Take one from each pair:

$$2, 2, 3$$

• Multiply:

$$2 \times 2 \times 3 = 12$$

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

Comparing Squares and Square Roots

Understanding the difference between squares and square roots is crucial. Here's a comparison:

This table helps Class 8 students quickly recall the concepts during exams.

Worked Example: Finding Square and Square Root

Example 1: Find the square of 15.

Solution:

$$15^2 = 15 \times 15 = 225$$

Example 2: Find the square root of 196 using prime factorization.

Solution:

• Prime factors of 196:

$$196 = 2 \times 2 \times 7 \times 7$$

• Pair factors:

$$(2 \times 2), (7 \times 7)$$

• Take one from each pair:

$$2, 7$$

• Multiply:

$$2 \times 7 = 14$$

So, $\sqrt{196} = 14$.

These examples show simple ways to calculate squares and square roots.

Importance of Squares and Square Roots in Class 8 Maths

Squares and square roots are fundamental concepts in the Class 8 NCERT Mathematics syllabus. They are used in:

• Algebraic expressions and equations
• Geometry (calculating areas and side lengths)
• Number theory
• Solving real-life problems involving measurements

Mastering these concepts helps students build a strong foundation for higher classes and competitive exams. Practising problems on squares and square roots improves calculation speed and accuracy.