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Class 8 — Mathematics (NCERT)

Squares and Square Roots

Class 8

  • ✓By the end of this lesson students will be able to understand what a perfect square number is.
  • ✓By the end of this lesson students will be able to identify and apply various properties of square numbers.
  • ✓By the end of this lesson students will be able to find the square root of a number using different methods.
  • ✓By the end of this lesson students will be able to solve problems involving squares and square roots.

Key concepts

Square of a Number

When a number is multiplied by itself, the product obtained is called the square of that number. For example, the square of 4 is 4 × 4 = 16. We write this as 4² = 16.

n × n = n²
Perfect Square (or Square Number)

A natural number is called a perfect square or a square number if it is the square of some natural number. For example, 1, 4, 9, 16, 25, 36, ... are perfect squares because they are squares of 1, 2, 3, 4, 5, 6, ... respectively.

n = m², where n and m are natural numbers
Properties of Square Numbers

1. Numbers ending in 2, 3, 7, or 8 are never perfect squares.\n2. Numbers ending in 0, 1, 4, 5, 6, or 9 might be perfect squares.\n3. If a number has 0 at its unit's place, its square ends with an even number of zeroes (e.g., 10² = 100, 20² = 400).\n4. The square of an even number is always an even number.\n5. The square of an odd number is always an odd number.\n6. The unit's digit of the square of a number depends on the unit's digit of the original number (e.g., if a number ends in 1 or 9, its square ends in 1).\n7. There are 2n non-perfect square numbers between n² and (n+1)².\n8. The sum of the first 'n' odd natural numbers is n² (e.g., 1+3+5 = 9 = 3²).

Square Root

The square root is the inverse operation of squaring. If m² = n, then m is the square root of n. For example, since 5² = 25, the square root of 25 is 5. We denote the square root using the symbol '√'. So, √25 = 5. Every positive perfect square number has two square roots, one positive and one negative. For Class 8, we generally consider the positive square root.

If m² = n, then m = √n
Methods to Find Square Roots

1. **Repeated Subtraction Method**: This method involves subtracting consecutive odd numbers (1, 3, 5, 7, ...) from the given number until the result is 0. The number of subtractions performed gives the square root.\n2. **Prime Factorisation Method**: This method involves finding the prime factors of the given number. Then, group the identical prime factors in pairs. For each pair, take one factor. The product of these chosen factors is the square root of the number.\n3. **Division Method**: This is a general method suitable for finding the square root of larger numbers or numbers that are not perfect squares (though for Class 8, we mostly deal with perfect squares). It involves grouping digits in pairs from the right, then systematically dividing and finding the square root.

Key facts to remember

  • 1A number ending in 2, 3, 7, or 8 is never a perfect square.
  • 2The square of an even number is always even, and the square of an odd number is always odd.
  • 3The number of zeroes at the end of a perfect square is always even.
  • 4The sum of the first 'n' odd natural numbers is equal to n².
  • 5The square root is the inverse operation of squaring a number.
  • 6The symbol '√' is used to denote the positive square root of a number.
  • 7Between any two consecutive perfect squares n² and (n+1)², there are 2n non-perfect square numbers.

Worked examples

Example 1

Without doing any calculation, find out which of the following numbers are not perfect squares:\n(i) 153\n(ii) 257\n(iii) 408\n(iv) 441

IWe know that numbers ending in 2, 3, 7, or 8 are never perfect squares.
IIFor (i) 153: The unit's digit is 3. Hence, 153 is not a perfect square.
IIIFor (ii) 257: The unit's digit is 7. Hence, 257 is not a perfect square.
IVFor (iii) 408: The unit's digit is 8. Hence, 408 is not a perfect square.
VFor (iv) 441: The unit's digit is 1. This number might be a perfect square (in fact, 21² = 441).

Answer

The numbers 153, 257, and 408 are not perfect squares.

This property helps in quickly identifying non-perfect squares.

Example 2

Find the square root of 729 by the Prime Factorisation Method.

IStep 1: Find the prime factors of 729.
II729 ÷ 3 = 243
III243 ÷ 3 = 81
IV81 ÷ 3 = 27
V27 ÷ 3 = 9
VI9 ÷ 3 = 3
VII3 ÷ 3 = 1
VIIISo, the prime factorisation of 729 is 3 × 3 × 3 × 3 × 3 × 3.
9Step 2: Group the prime factors in pairs.
10729 = (3 × 3) × (3 × 3) × (3 × 3)
11Step 3: Take one factor from each pair and multiply them.
12√729 = 3 × 3 × 3
13√729 = 27

Answer

The square root of 729 is 27.

Ensure all prime factors are correctly identified and paired.

Example 3

Find the square root of 4096 by the Division Method.

IStep 1: Place a bar over every pair of digits starting from the unit's digit. If the number of digits is odd, the leftmost single digit will also have a bar.
II _ _
III √40 96
IVStep 2: Find the largest number whose square is less than or equal to the first group (40). 6² = 36, 7² = 49. So, 6 is the number. Write 6 as the quotient and also as the divisor.
V 6
VI ____
VII6 |√40 96
VIII -36
9 ____
10 4
11Step 3: Bring down the next pair of digits (96) to the right of the remainder (4). The new dividend is 496.
12 6
13 ____
146 |√40 96
15 -36
16 ____
17 4 96
18Step 4: Double the quotient (6 × 2 = 12) and write it with a blank digit to its right. This forms the new divisor. We need to find a digit (let's call it 'x') such that when 12x is multiplied by x, the product is less than or equal to 496.
19 6 x
20 ____
216 |√40 96
22 -36
23 ____
2412x| 4 96
25Try x=4: 124 × 4 = 496.
26Step 5: Write 4 as the next digit in the quotient and also in the divisor. Subtract the product.
27 6 4
28 ____
296 |√40 96
30 -36
31 ____
32124| 4 96
33 -4 96
34 ____
35 0
36Step 6: Since the remainder is 0 and there are no more pairs to bring down, the process stops. The quotient is the square root.

Answer

The square root of 4096 is 64.

Practice is key for mastering the division method. Always remember to double the quotient for the next step's divisor.

Common mistakes

  • ✗Confusing the square of a number with its square root (e.g., thinking the square of 4 is 2, instead of 16).
  • ✗Incorrectly grouping digits in the division method, especially when the number of digits is odd.
  • ✗Making errors in prime factorisation, leading to incorrect square roots.
  • ✗Assuming a number is a perfect square just because its unit digit is 0, 1, 4, 5, 6, or 9, without further verification.
  • ✗Forgetting to pair identical prime factors in the prime factorisation method, or taking only one factor from a pair.

Exam tips

  • ★Memorise the squares of natural numbers up to at least 20 to quickly solve problems.
  • ★Practice all three methods (repeated subtraction, prime factorisation, and division method) for finding square roots thoroughly.
  • ★Always check your answer by squaring the obtained square root to ensure it matches the original number.
  • ★Pay attention to the unit digit of the given number to quickly eliminate options or identify non-perfect squares.
  • ★Show all steps clearly in worked examples, especially for the prime factorisation and division methods, as marks are often awarded for correct method.

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