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Squares and Square Roots

    Exercise 5.3

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8th class > Squares and Square Roots > Exercise 5.3

Exercise 5.3

  1. What could be the possible ‘one’s’ digits of the square root of each of the following numbers?

Instructions

(i) 9801
Considering the one's digit of 9801, which is .
The possible one's digits of a number whose square ends in 1 are the numbers that when squared, their result ends in 1. We know: 12 = , 92 =
Therefore, the possible one's digits of the square root of 9801 are and . (Enter digits in ascending order).
(ii) 99856
Considering the one's digit of 99856, which is . We know: 42 = , 62 =
Therefore, the possible one's digits of the square root of 99856 are and . (Enter digits in ascending order).
(iii) 998001
Considering the one's digit of 998001, which is . We know: 12 = , 92 =
Therefore, the possible one's digits of the square root of 998001 are and . (Enter digits in ascending order).
(iv) 657666025
Considering the one's digit of 998001, which is . We know: 52 = .
Therefore, the possible one's digit of the square root of 657666025 is .

Instructions

  1. Without doing any calculation, find the numbers which are surely not perfect squares.
(i) 153
Taking the number 153 which has a unit digit of . A perfect square end in 3.
Therefore, 153 is surely not a perfect square.
(ii) 257
Taking the number 257 which has a unit digit of . A perfect square end in 7.
Therefore, 257 is surely not a perfect square.
(iii) 408
Taking the number 408 which has a unit digit of . A perfect square end in 8.
Therefore, 408 is surely not a perfect square.
(iv) 441
Taking the number 441 which has a unit digit of . A perfect square end in 1.
Therefore, 441 can be a perfect square.
  1. Find the square roots of 100 and 169 by the method of repeated subtraction:

(i) 100

Instructions

Start the subtraction operation from 100

  • Start with 100: Subtract 1 (1st odd number): 100 - 1 =
  • Subtract 3 (2nd odd number): 99 - 3 =
  • Subtract 5 (3rd odd number): 96 - 5 =
  • Subtract 7 (4th odd number): 91 - 7 =
  • Subtract 9 (5th odd number): 84 - 9 =
  • Subtract 11 (6th odd number): 75 - 11 =
  • Subtract 13 (7th odd number): 64 - 13 =
  • Subtract 15 (8th odd number): 51 - 15 =
  • Subtract 17 (9th odd number): 36 - 17 =
  • Subtract 19 (10th odd number): 19 - 19 =
  • We have found the answer. The square root of 100 is 10.

(ii) 169

Instructions

Start the subtraction operation from 169

  • Start with 169: Subtract 1 (1st odd number): 169 - 1 =
  • Subtract 3 (2nd odd number): 168 - 3 =
  • Subtract 5 (3rd odd number): 165 - 5 =
  • Subtract 7 (4th odd number): 160 - 7 =
  • Subtract 9 (5th odd number): 153 - 9 =
  • Subtract 11 (6th odd number): 144 - 11 =
  • Subtract 13 (7th odd number): 133 - 13 =
  • Subtract 15 (8th odd number): 120 - 15 =
  • Subtract 17 (9th odd number): 105 - 17 =
  • Subtract 19 (10th odd number): 88 - 19 =
  • Subtract 21 (11th odd number): 69 - 21 =
  • Subtract 23 (12th odd number): 48 - 23 =
  • Subtract 25 (13th odd number): 25 - 25 =
  • We have found the answer. The square root of 169 is 13.
  1. Find the square roots of the following numbers by the Prime Factorisation Method.

Instructions

(i) 729 = × × × × ×
Thus, 729 = × × =
(ii) 400 = × × × × ×
Thus, 400 = × × =
(iii) 1764 = × × × × ×
Thus, 1764 = × × =
(iv) 4096 = × × × × × × × × × × ×
Thus, 4096 = × × × × × =
(v) 7744 = × × × × × × ×
Thus, 7744 = × × × =
(vi) 9604 = × × × × ×
Thus, 9604 = × × =
(vii) 5929 = × × ×
Thus, 5929 = × =
(viii) 9216 = × × × × × × × × × × ×
Thus, 9216 = × × × × × =
(ix) 529 = ×
Thus, 529 =
(x) 8100 = × × × × × × ×
Thus, 8100 = × × × =
  1. For each of the following numbers, find the smallest whole number by which it should be multiplied so as to get a perfect square number. Also find the square root of the square number so obtained.

Instructions

(i) 252 = × × × ×
Thus, is the smallest whole number by which the given number is multiplied to get a perfect square number (as it doesn't have a pair). The new square number becomes 252 × =
Square root of 1764: 1764 = × × =
(ii) 180 = × × × ×
Thus, is the smallest whole number by which the given number is multiplied to get a perfect square number (as it doesn't have a pair). The new square number becomes 180 × =
Square root of 900: 900 = × × =
(iii) 1008 = × × × × × ×
Thus, is the smallest whole number by which the given number is multiplied to get a perfect square number (as it doesn't have a pair). The new square number becomes 1008 × =
Square root of 7056: 7056 = × × × =
(iv) 2028 = × × × ×
Thus, is the smallest whole number by which the given number is multiplied to get a perfect square number (as it doesn't have a pair). The new square number becomes 2028 × =
Square root of 6084: 6084 = × × =
(v) 1458 = × × × × × ×
Thus, is the smallest whole number by which the given number is multiplied to get a perfect square number (as it doesn't have a pair). The new square number becomes 1458 × =
Square root of 2916: 2916 = × × × =
(vi) 768 = × × × × × × × ×
Thus, is the smallest whole number by which the given number is multiplied to get a perfect square number (as it doesn't have a pair). The new square number becomes 768 × =
Square root of 2304: 2304 = × × × × =
  1. For each of the following numbers, find the smallest whole number by which it should be divided so as to get a perfect square. Also find the square root of the square number so obtained.

Instructions

(i) 252 = × × × ×
Thus, is the smallest whole number by which the given number is divided to get a perfect square number (as it doesn't have a pair).The new square number becomes 252 =
Square root of 36: 36 = × =
(ii) 2925 = × × × ×
Thus, is the smallest whole number by which the given number is divided to get a perfect square number (as it doesn't have a pair). The new square number becomes 2925 =
Square root of 225: 225 = × =
(iii) 396 = × × × ×
Thus, is the smallest whole number by which the given number is divided to get a perfect square number (as it doesn't have a pair). The new square number becomes 396 =
Square root of 36: 36 = × =
(iv) 2645 = × ×
Thus, is the smallest whole number by which the given number is divided to get a perfect square number (as it doesn't have a pair).The new square number becomes 2645 =
Square root of 529: 529 = × =
(v) 2800 = × × × × × ×
Thus, is the smallest whole number by which the given number is divided to get a perfect square number (as it doesn't have a pair). The new square number becomes 2800 =
Square root of 400: 400 = × × =
(vi) 1620 = × × × × × ×
Thus, is the smallest whole number by which the given number is divided to get a perfect square number (as it doesn't have a pair). The new square number becomes 1620 =
Square root of 324: 324 = × × × × × =
  1. The students of Class VIII of a school donated Rs. 2401 in all, for Prime Minister’s National Relief Fund. Each student donated as many rupees as the number of students in the class. Find the number of students in the class.

Instructions

Let the number of students in the class be n. Each student donated n rupees.
The total amount donated by the students is Rs. 2401.
Therefore, the required equation becomes =
We need to find the value of n that satisfies the equation.
Looking at the unit digit, we see that the number 2401 a perfect square.
Prime factorisation of 2401 = × × ×
Upon calculating, we find that the square root of 2401 is .
Therefore, the number of students in the class is 49.
  1. 2025 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row.

Instructions

Let the number of plants in each row be t. Each row has t plants arranged along it.
The total number of plants is 2025.
Therefore, the required equation becomes =
We need to find the value of t that satisfies the equation.
Looking at the unit digit, we see that the number 2025 a perfect square.
Prime factorisation of 2025 = × × × × ×
Upon calculating, we find that the square root of 2025 is .
Therefore, the number of rows and plants per row is 45.
  1. Find the smallest square number that is divisible by each of the numbers 4, 9 and 10.

Instructions

To find the smallest square number divisible by 4, 9 and 10, we need to find the of the three numbers.
Upon calculating: LCM of 4, 9, 10 =
Prime factorization of 180 = × × × × (Enter in ascending order).
Since, has , it is the number that needs to be to get the smallest square number.
The required smallest square number = 180 × 5 =
So, 900 is the smallest square number that is divisible by 4, 9 and 10.
  1. Find the smallest square number that is divisible by each of the numbers 8, 15 and 20.

Instructions

To find the smallest square number divisible by 8, 15 and 20, we need to find the of the three numbers.
Upon calculating: LCM of 8, 15, 20 =
Prime factorization of 120 = × × × × (Enter in ascending order).
Since, 2, 3 and 5 have , they need to be to get the smallest square number.
The required smallest square number = 120 × 2 × 3 × 5 =
So, 3600 is the smallest square number that is divisible by 8, 15 and 20.