Class 8 Notes

Class 8 Mathematics Notes Chapter 6 (Squares and Square Roots) – Mathematics Book

Philoid

Philoid

— 7 min read

Mathematics
Alright class, let's focus on Chapter 6: Squares and Square Roots. This is a fundamental chapter, and understanding it well is crucial not just for your class exams but also for various competitive government exams where quantitative aptitude is tested. Pay close attention to the properties and methods.

Chapter 6: Squares and Square Roots - Detailed Notes

1. Introduction to Squares

  • Definition: When a number is multiplied by itself, the product obtained is called the square of that number.
    • Example: The square of 3 is 3 × 3 = 9. We write this as 3² = 9.
    • Example: The square of 12 is 12 × 12 = 144. We write this as 12² = 144.
  • Perfect Square (or Square Number): A natural number is called a perfect square if it is the square of some natural number.
    • Examples: 1 (1²), 4 (2²), 9 (3²), 16 (4²), 25 (5²), 36 (6²), ... are perfect squares.
    • Numbers like 2, 3, 5, 6, 7, 8, 10, 11, ... are not perfect squares.

2. Properties of Square Numbers (Very Important for Exams)

  • (i) Units Digit: A number ending in 2, 3, 7, or 8 is never a perfect square.
    • Exam Tip: This helps quickly eliminate options in MCQs.
  • (ii) Units Digit: A number ending in 0, 1, 4, 5, 6, or 9 may or may not be a perfect square.
    • Example: 100 (ends in 0, is 10²), 81 (ends in 1, is 9²), 64 (ends in 4, is 8²), 25 (ends in 5, is 5²), 36 (ends in 6, is 6²), 49 (ends in 9, is 7²).
    • Counter-examples: 10, 19, 24, 35, 46 are not perfect squares.
  • (iii) Even/Odd Squares:
    • The square of an even number is always even. (e.g., 6² = 36)
    • The square of an odd number is always odd. (e.g., 7² = 49)
  • (iv) Zeros at the End: A number ending in an odd number of zeros is never a perfect square. A perfect square always has an even number of zeros at the end.
    • Example: 100 (2 zeros, 10²), 40000 (4 zeros, 200²).
    • Example: 10, 1000, 900000 are not perfect squares.
  • (v) Relation between Squares of Consecutive Numbers: The difference between the squares of two consecutive natural numbers n and (n+1) is equal to their sum.
    • (n+1)² - n² = (n+1 + n) = 2n + 1
    • Example: 5² - 4² = 25 - 16 = 9. Also, 2(4) + 1 = 8 + 1 = 9.
    • Application: This tells us there are 2n non-perfect square numbers between the squares of consecutive numbers n and (n+1). (Between 4²=16 and 5²=25, there are 2*4 = 8 numbers: 17, 18, 19, 20, 21, 22, 23, 24).
  • (vi) Sum of Odd Numbers: The sum of the first n odd natural numbers is n².
    • 1 = 1 = 1²
    • 1 + 3 = 4 = 2²
    • 1 + 3 + 5 = 9 = 3²
    • 1 + 3 + 5 + 7 = 16 = 4²
  • (vii) Pythagorean Triplets: Three natural numbers m, n, p are said to form a Pythagorean triplet if m² + n² = p².
    • For any natural number m > 1, the triplet (2m, m² - 1, m² + 1) forms a Pythagorean triplet.
    • Example: If m = 3, then 2m = 6, m² - 1 = 3² - 1 = 8, m² + 1 = 3² + 1 = 10. So, (6, 8, 10) is a Pythagorean triplet (6² + 8² = 36 + 64 = 100 = 10²).
    • Exam Tip: Often, one number of the triplet is given, and you need to find the others using these forms. Check if the given number fits 2m, m²-1, or m²+1.

3. Finding the Square of a Number

  • Direct Multiplication: Multiply the number by itself. (Suitable for small numbers).
  • Using Identities: For larger numbers, algebraic identities can simplify calculation.
    • (a + b)² = a² + 2ab + b²
      • Example: 43² = (40 + 3)² = 40² + 2(40)(3) + 3² = 1600 + 240 + 9 = 1849
    • (a - b)² = a² - 2ab + b²
      • Example: 98² = (100 - 2)² = 100² - 2(100)(2) + 2² = 10000 - 400 + 4 = 9604
  • Special Case (Numbers ending in 5): For a number ending in 5, say n5. Its square ends in 25, and the preceding digits are given by n(n+1).
    • Example: 35² = ? Here n=3. n(n+1) = 3(4) = 12. So, 35² = 1225.
    • Example: 75² = ? Here n=7. n(n+1) = 7(8) = 56. So, 75² = 5625.

4. Introduction to Square Roots

  • Definition: Finding the square root is the inverse operation of squaring. If m is a perfect square, say m = n², then n is the square root of m.
  • Symbol: The symbol for square root is √. So, √m = n.
    • Example: Since 6² = 36, the square root of 36 is 6 (√36 = 6).
    • Note: Technically, √36 = ±6, but in this context (NCERT Class 8), we usually consider the positive square root (principal square root).

5. Methods for Finding Square Roots

  • (i) Repeated Subtraction:
    • Subtract successive odd numbers (1, 3, 5, 7, ...) from the given perfect square until you reach 0.
    • The number of steps taken is the square root of the number.
    • Example: Find √81
      • 81 - 1 = 80 (Step 1)
      • 80 - 3 = 77 (Step 2)
      • 77 - 5 = 72 (Step 3)
      • 72 - 7 = 65 (Step 4)
      • 65 - 9 = 56 (Step 5)
      • 56 - 11 = 45 (Step 6)
      • 45 - 13 = 32 (Step 7)
      • 32 - 15 = 17 (Step 8)
      • 17 - 17 = 0 (Step 9)
      • Since we reached 0 in 9 steps, √81 = 9.
    • Limitation: Only practical for small perfect squares.
  • (ii) Prime Factorization Method:
    • Step 1: Find the prime factors of the given number.
    • Step 2: Group the factors in pairs of equal numbers.
    • Step 3: Take one factor from each pair.
    • Step 4: Multiply these chosen factors to get the square root.
    • Example: Find √784
      • Prime Factorization: 784 = 2 × 2 × 2 × 2 × 7 × 7
      • Pairing: 784 = (2 × 2) × (2 × 2) × (7 × 7)
      • Taking one from each pair: 2 × 2 × 7
      • Multiplication: 2 × 2 × 7 = 28. So, √784 = 28.
    • Application: Useful for determining if a number is a perfect square (if all factors form pairs). Also used to find the smallest number to multiply or divide to make a given number a perfect square.
      • Example: Find the smallest number to multiply 252 by to get a perfect square.
        • 252 = 2 × 2 × 3 × 3 × 7 = (2 × 2) × (3 × 3) × 7
        • The factor 7 is unpaired. So, multiply by 7. 252 × 7 = 1764 = (2×2)×(3×3)×(7×7). √1764 = 2×3×7 = 42.
  • (iii) Division Method (Long Division): (Most versatile method, works for large numbers and decimals)
    • Step 1: Place bars over pairs of digits starting from the units place (e.g., 7 84, 5 29). For the leftmost digit(s) if single, treat it as a pair.
    • Step 2: Find the largest number whose square is less than or equal to the number under the leftmost bar. Take this number as the divisor and the quotient.
    • Step 3: Subtract the product of the divisor and quotient from the number under the leftmost bar.
    • Step 4: Bring down the number under the next bar to the right of the remainder. This becomes the new dividend.
    • Step 5: Double the current quotient and write it down with a blank space on its right.
    • Step 6: Guess the largest possible digit to fill the blank, which will also be the next digit of the quotient. The product of this new digit and the new divisor (formed in Step 5 + blank filled) should be less than or equal to the new dividend.
    • Step 7: Subtract this product from the dividend.
    • Step 8: Repeat steps 4-7 until all bars have been considered and the remainder is 0 (for perfect squares).
    • Example: Find √529
            2  3
    _______
   2| 5 29
    | -4
    |_____
  43| 1 29
    | -1 29
    |______
    |    0
      • Pairing: 5 29
      • Largest square ≤ 5 is 2²=4. Divisor=2, Quotient=2. Subtract 4 from 5, remainder=1.
      • Bring down 29. New dividend=129.
      • Double the quotient (2): 2*2=4. New divisor is 4_.
      • Guess digit for blank: 42×2=84, 43×3=129. So, the digit is 3. New quotient digit=3.
      • Subtract 129 from 129. Remainder=0.
      • So, √529 = 23.

6. Square Roots of Decimals

  • Use the division method.
  • Place bars on the integral part from right to left.
  • Place bars on the decimal part from left to right. Add a zero at the end if needed to make pairs.
  • Place the decimal point in the quotient as soon as the integral part is exhausted.
  • Example: Find √17.64
          4 . 2
    _______
   4| 17 . 64
    | -16
    |______
  82|  1  64
    | -1  64
    |_______
    |     0
    • Pairing: 17 . 64
    • Largest square ≤ 17 is 4²=16. Divisor=4, Quotient=4. Remainder=1.
    • Place decimal in quotient. Bring down 64. New dividend=164.
    • Double quotient (4): 2*4=8. New divisor is 8_.
    • Guess digit: 82 × 2 = 164. So, the digit is 2. New quotient digit=2.
    • Subtract 164. Remainder=0.
    • So, √17.64 = 4.2.

7. Estimating Square Roots

  • For non-perfect squares, we can estimate the square root.
  • Example: Estimate √80
    • We know 8² = 64 and 9² = 81.
    • Since 80 is very close to 81, √80 will be slightly less than √81 = 9.
    • A reasonable estimate is ≈ 8.9. (Actual value is approx 8.944).

8. Typical Exam Problems

  • Identifying perfect squares based on properties.
  • Finding the smallest number to multiply/divide to get a perfect square.
  • Finding the smallest/largest number of 'n' digits which is a perfect square.
  • Finding square roots using prime factorization or division method.
  • Finding square roots of decimals or fractions.
  • Problems involving Pythagorean triplets.
  • Word problems involving area of squares (Area = side², Side = √Area) or arrangement in rows/columns (Total items = rows × columns; if rows = columns, Total = rows²).

Multiple Choice Questions (MCQs)

  1. Which of the following numbers is NOT a perfect square?
    (a) 121
    (b) 144
    (c) 169
    (d) 153

  2. The square root of 1764 found by prime factorization is:
    (a) 32
    (b) 42
    (c) 48
    (d) 52

  3. What is the smallest number by which 392 must be multiplied so that the product is a perfect square?
    (a) 2
    (b) 3
    (c) 7
    (d) 5

  4. A number ending in which digit can never be a perfect square?
    (a) 1
    (b) 5
    (c) 0
    (d) 8

  5. Find the value of √1.96.
    (a) 1.4
    (b) 1.6
    (c) 0.14
    (d) 14

  6. If one member of a Pythagorean triplet is 10 (using the form 2m, m²-1, m²+1), what are the other two members?
    (a) 6, 8
    (b) 24, 26
    (c) 99, 101
    (d) Cannot be determined uniquely

  7. How many non-perfect square numbers lie between 11² and 12²?
    (a) 21
    (b) 22
    (c) 23
    (d) 24

  8. The square root of 6084 using the division method is:
    (a) 72
    (b) 78
    (c) 82
    (d) 88

  9. A General wishing to arrange his men, who were 6000 in number, in the form of a square found that he had 71 men left over. How many men were in the front row?
    (a) 73
    (b) 77
    (c) 87
    (d) 79

  10. The value of √(248 + √52 + √144) is:
    (a) 14
    (b) 16
    (c) 18
    (d) 20

Answer Key for MCQs:

  1. (d) 153 (ends in 3)
  2. (b) 42 (1764 = 2x2 x 3x3 x 7x7 = (2x3x7)²)
  3. (a) 2 (392 = 2x2x2 x 7x7. Needs one more 2 for pairing)
  4. (d) 8 (Numbers ending in 2, 3, 7, 8 are never perfect squares)
  5. (a) 1.4 (√196 = 14, so √1.96 = 1.4)
  6. (b) 24, 26 (If 2m = 10, m=5. Then m²-1 = 24, m²+1 = 26)
  7. (b) 22 (Between n² and (n+1)², there are 2n non-squares. Here n=11, so 2*11 = 22)
  8. (b) 78 (Using division method)
  9. (b) 77 (Number of men arranged = 6000 - 71 = 5929. Number in front row = √5929 = 77)
  10. (b) 16 (√144 = 12; √52+12 = √64 = 8; √248+8 = √256 = 16)

Make sure you practice finding squares and square roots, especially using the prime factorization and division methods. Remember those properties – they are shortcuts in disguise for competitive exams! Let me know if any part needs further clarification.

Read more