Squares and Square Roots Class 8 Maths Formulas

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Squares and Square Roots Class 8 Maths Formulas

Square Number: The square of a number is the product of the number with the number itself The, square of x = (x × x), denoted by x 2.

A natural number n is a perfect square, if n = m² for some natural number m.{1 = 1 × 1 = 1², 4 = 2 × 2 = 2²}

Square Root: Square root is the inverse operation of square, i.e., the positive square root of a number is denoted by the symbol $\sqrt{ }$ 

For example, $3^{2}=9$ gives $\sqrt{9}=3$ or $\left(3^{2}\right)^{1 / 2}=3 .$

For positive numbers $a$ and $b$, we have

(i) $\sqrt{a b}=\sqrt{a} \times \sqrt{b}$,

(ii) $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ or $\sqrt{\frac{a}{b}}=\left(\frac{a}{b}\right)^{\frac{1}{2}}$

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What Are Square Numbers and Perfect Squares?

Square of a number is the product of that number multiplied by itself.

Square of x = x × x = x²

A natural number n is called a perfect square if n = m² for some natural number m.

Examples of Perfect Squares

• 1 = 1 × 1 = 1²
• 4 = 2 × 2 = 2²
• 9 = 3 × 3 = 3²
• 16 = 4 × 4 = 4²
• 25 = 5 × 5 = 5²

How to Identify If a Number Is a Perfect Square

A number is a perfect square only when:

• Every prime factor in its prime factorisation appears an even number of times
• It does not end in 2, 3, 7, or 8
• It does not end in an odd number of zeros

Square Root — Definition and Core Formulas

The square root is the inverse operation of squaring. The positive square root of a number n is written as √n.

If 3² = 9, then √9 = 3

Core Square Root Formulas

For positive numbers a and b:

The product and division rules for square roots are the most frequently tested formulas in Class 8 exams. Practise splitting larger numbers — for example, √(36 × 25) = √36 × √25 = 6 × 5 = 30. This shortcut saves valuable time in exams and becomes even more useful in JEE-level simplification problems.

Properties of Square Numbers

Understanding these properties helps you instantly identify perfect squares — a skill that saves time in both school tests and competitive exams.

Properties Related to the Unit Digit

• A number ending in 2, 3, 7, or 8 is never a perfect square
• A number ending in an odd number of zeros is never a perfect square
• A perfect square ending in 1 has a square root ending in 1 or 9
• A perfect square ending in 4 has a square root ending in 2 or 8
• A perfect square ending in 9 has a square root ending in 3 or 7
• A perfect square ending in 6 has a square root ending in 4 or 6
• A perfect square ending in 5 has a square root ending in 5

Properties Related to Even and Odd Numbers

• The square of an even number is always even
• The square of an odd number is always odd
• The square of a proper fraction is always smaller than the original fraction

Pattern Properties

• For every natural number n: (n+1)² − n² = (n+1) + n
• Sum of the first n odd natural numbers = n²
  • Example: 1 + 3 + 5 + 7 = 4² = 16
• There are 2n non-perfect-square numbers between n² and (n+1)²

Perfect Squares from 1 to 20 — Quick Reference Table

Memorising squares up to 20 is strongly recommended — it directly helps in factorisation, geometry, and CBSE board questions.

What Are Pythagorean Triplets?

A Pythagorean triplet is a set of three natural numbers (m, n, p) such that:

m² + n² = p²

The most common example is (3, 4, 5) because 9 + 16 = 25.

Formula to Generate Pythagorean Triplets

For every natural number m > 1:

(2m, m² − 1, m² + 1) is a Pythagorean triplet

Examples

Pythagorean triplets link Chapter 5 directly to geometry and, later, trigonometry. Students preparing for JEE from Class 8 should note that the triplet (3, 4, 5) and its multiples appear constantly in coordinate geometry and right-triangle problems. Building this pattern recognition early makes those problems effortless later.

How Do You Find the Square Root of a Number?

Two standard methods are taught in CBSE Class 8 for finding square roots.

Method 1 — Prime Factorisation

Steps:

1. Write the number as a product of its prime factors
2. Group the prime factors in pairs
3. Take one factor from each pair
4. Multiply the selected factors — the result is the square root

Example: √324

• 324 = 2 × 2 × 3 × 3 × 3 × 3
• Pairs: (2, 2), (3, 3), (3, 3)
• √324 = 2 × 3 × 3 = 18

Method 2 — Long Division Method

The long division method is used for larger numbers or decimals where prime factorisation is not practical.

Steps:

1. Group digits in pairs from the decimal point outward (right to left for whole number part)
2. Find the largest integer whose square ≤ the first group
3. Subtract, bring down the next pair
4. Double the current quotient to form the new divisor, find the next digit
5. Repeat until the required precision is reached

Example: √625 = 25 (verified: 25 × 25 = 625)

For detailed step-by-step NCERT solutions on this method, visit the NCERT Solutions for Class 8 Maths page on eSaral.

Worked Examples

Example 1 — Using the Sum of Odd Numbers Property

Q: Express 49 as a sum of consecutive odd numbers.

• 49 = 7² = 1 + 3 + 5 + 7 + 9 + 11 + 13 ✓ (7 odd numbers)

Example 2 — Identifying a Perfect Square

Q: Is 2028 a perfect square?

• 2028 ends in 8 → A number ending in 8 is never a perfect square.
• Answer: No.

Example 3 — Pythagorean Triplet

Q: Find the Pythagorean triplet with m = 6.

• 2m = 12, m² − 1 = 35, m² + 1 = 37
• Triplet: (12, 35, 37)
• Check: 144 + 1225 = 1369 = 37² ✓

Example 4 — Applying the Square Root Division Rule

Q: Simplify √(144/169)

• √144 / √169 = 12/13

For more practice, explore NCERT Solutions for Class 11 Maths and NCERT Solutions for Class 12 Maths to see how square root concepts extend into higher-level algebra.

• A number ending in 2, 3, 7 or 8 is never a perfect square.
• A number ending in an odd number of zeros is never a perfect square.
• The square of an even number is even.
• The square of an odd number is odd.
• The square of a proper fraction is smaller than the fraction.
• For every natural number $n$, we have $\left\{(n+1)^{2}-n^{2}\right\}=\{(n+1)+n\}$.
• Sum of first $n$ odd natural numbers $=n^{2}$.
• If $m, n, p$ are natural numbers such that $\left(m^{2}+n^{2}\right)=p^{2}$, then $(m, n, p)$ is called a Pythagorean triplet.
• For every natural number $m>1,\left(2 m, m^{2}-1, m^{2}+1\right)$ is a Pythagorean triplet.
• There are $2 n$ non-perfect square numbers between the squares of the number $n$ and $(n+1)$
• The numbers which can be expressed as the product of the number with itself are called square numbers or perfect squares. For example, $1,4,9,16,25, \ldots .$
• If a natural number $m$ can be expressed as $n^{2}$, where $n$ is also a natural number, then, $m$ is called a square number.

Also Read,

NCERT Class 8 Maths Book Download

NCERT Class 8 Science Book Download

NCERT Class 8 Science Exemplar Download

NCERT Class 8 Maths Exemplar Download

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