Class 8 Maths Chapter 6 Notes - Squares and Square Roots

Square Numbers

The term ‘square number’ means a number multiplied twice to get the square. The term ‘square number’ came with the concept of the area of a square. Just like the area of a square means the product of two equal sides, the square number is the product of the same number multiplied twice.
Let’s have a look at the properties of the square numbers.

• The squares of odd numbers are odd, while that of even numbers are even.
• Square numbers ends with 1, 3, 4, 5, 6, or 9 at its units place.
• Numbers ending with 2, 3, 7, or 8 are not square numbers.
• If a number has n zeroes at the end, its square will have 2n zeroes at the end.

Square numbers follow some patterns.

• On adding two consecutive triangular numbers, we get a square number. Triangular numbers are numbers whose dot patterns can be arranged as squares.
• In between the squares of two consecutive numbers n and (n + 1), there are 2n non perfect square numbers.
• The sum of first n odd natural numbers is n2.
• Every square number can be written as the sum of two consecutive natural numbers.
• The product of two consecutive odd and even numbers (n + 1) and (n – 1) is n2 – 1.
Let us see how we can find square of a number.

Finding Square Numbers

Square of a number can be found using column method, diagonal method, visual method, or by using identities. Let us see the identities method.

Identity Method: Two main identities used to find square of a number are:

(a + b)2 = a2 + 2ab + b2
(a – b)2 = a2 – 2ab + b2

Let us see how we can find square of 72 using identities.

722
= (70 + 2)2
= 702 + 2(70)(2) + 22
= 4900 + 280 + 4
= 5184

Pythagorean Triplets

If the sum of two square numbers is a square number, then these three numbers form a Pythagorean triplet. For any number n > 1, we have (2n) 2 + (n2 − 1)2 = (n2 + 1)2. So, 2n, n2 − 1 and n2 + 1 forms a Pythagorean triplet.

Let us read about square roots.

NCERT Solutions for Class 8 Maths Chapter 6
NCERT Solutions Class 8 Maths Exercise 6.1
NCERT Solutions Class 8 Maths Exercise 6.2
NCERT Solutions Class 8 Maths Exercise 6.3
NCERT Solutions Class 8 Maths Exercise 6.4

Square Roots

The square root of a number x is a number, which when multiplied with itself (squared), gives x as the answer. For example, the square root of 9 is 3, because 3 multiplied with itself, gives 9 as the answer.

Perfect Square

A number whose square root is a whole number. The first few perfect squares are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144.

There are two methods for finding the square root of perfect squares-

Repeated Subtraction

To obtain the square root by repeated subtraction, subtract successive odd numbers from the given number, starting from 1 till we get 0 as the result. In this method, we cannot skip any odd number and we should always start from 1. The square root of the required number will be the number of steps used to get 0.

Prime Factorisation

In this method, express the number as the product of its prime factors and make pairs of two same factors. From each pair, take one factor and find its product to get the square root.
If a number is not a perfect square, we can make it a perfect square using prime factorisation method by multiplying/dividing it by the required number.

Let use read the long division method now.

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Long Division method

The long division method is used to find the square root of large numbers. This does not mean that we cannot use the long division method to find the square root of small numbers. We can find the square root of any number using this method.

This method is also used to find the square root for decimals and non-perfect squares. In long division method, we make pairs of digits starting from the right.

Let us understand the steps of long division method from the example given below.



If a number is not a perfect square, we can make it a perfect square using long division method by adding/subtracting some number from that number.

To find the square root of a fraction, we can find the square root of the numerator and the denominator separately and then combine them to obtain the square root of the fraction. Let us see an example.

Find the square root of 324/441.

324 = 2 × 2 × 3 × 3 × 3 × 3
= (2 × 2) × (3 × 3) × (3 × 3)
= 22 × 32 × 32
= (2 × 3 × 3)2
= 182
√324 = 18
441 = 3 × 3 × 7 × 7
= (3 × 3) × (7 × 7)
= 32 × 72
= (3 × 7)2
= 212
√441 = 21

Links to Other Resources for Class 8 Maths

Other Class 8 Maths Related Links
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