Class 8: Square Roots & Cube Roots – Exercise 7A – ICSE / ISC / CBSE Mathematics Portal for K12 Students

Question 1: Find the square of each of the following numbers:

(i) $14 = 14$ (ii) $137$ (iii) $\frac{4}{17}$ (iv) $2$ $\frac{3}{4}$ (v) $0.01$ (vi) of $1.2$ (vii) $0.17$
(viii) $4.6$

Answer:

(i) Square of $14 = 14 \times 14 = 196$

(ii) Square of $137 = 137 \times 137 = 18769$

(iii) Square of $\displaystyle \frac{4}{17} =\frac{16}{289}$

(iv) Square of $\displaystyle 2 \frac{3}{4} = \frac{121}{16}$

(v) Square $0.01 = 0.01 \times 0.01 = 0.0001$

(vi) Square of $1.2 = 1.2 \times 1.2 = 1.44$

(vii) Square of $0.17 =0.17 \times 0.17 = 0.0289$

(viii) Square of $4.6 = 4.6 \times 4.6 = 21.16$

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Question 2: Using prime factorization method, find which of the following are perfect square numbers:

(i) $196$ (ii) $252$ (iii) $324$ (iv) $1225$ (v) $2916$ (vi) $3582$ (vii) $4489$

Note: A natural number is called a perfect square, if it is the square of some natural number

Answer:

(i) $196 = 14 \times 14$ (hence perfect square)

(ii) $252 = 2 \times 2 \times 7 \times 3 \times 3$ (not a perfect square)

(iii) $324 = 18 \times 18$ (hence perfect square)

(iv) $1225 = 35 \times 35$ (hence a perfect square)

(v) $2916 = 54 \times 54$ (hence perfect square)

(vi) $3582 = 2 \times 3 \times 3 \times 199$ (not a perfect square)

(vii) $4489 = 67 \times 67$ (hence a perfect square)

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Question 3: Which of the following numbers are squares of even numbers?
Note: The Square of an even number is always an even number.

$676$ , $1089$ , $5625$ , $729$ , $2304$ , $9216$

Answer:

$676$ (square of $26$ ) $2304$ (square of $48$ ) and $9216$ (square of $96$ ) are square of even numbers.

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Question 4: Using prime factorization method, find the square root of each of the following numbers:

(i) $441$ (ii) $784$ (iii) $3969$ (vi) $4900$ (v) $11025$ (vi) $30625$

Answer:

(i) $441 = 3 \times 7 \times 3 \times 7$ . $\therefore$ , square root of $441 = 3 \times 7 = 21$

(ii) $784 = 4 \times 7 \times 4 \times 7$ . Therefore, square root of $784 = 4 \times 7 = 28$

(iii) $3969 = 7 \times 9 \times 7 \times 9$ . Therefore, square root of $3969 = 7 \times 9 = 63$

(vi) $4900 = 7 \times 10 \times 7 \times 10$ . Therefore, square root of $4900 = 70$

(v) $11025 = 3 \times 7 \times 5 \times 3 \times 7 \times 5$ . Therefore, square root of $11025 = 3 \times 7 \times 5 = 105$

(vi) $30625 = 5 \times 5 \times 7 \times 5 \times 5 \times 7$ . Therefore, square root of $30625 = 5 \times 7 \times 5 = 175$

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Question 5: The students of a class arranged a picnic. Each student contributed as many rupees as the number of students in the class. If the total contribution is Rs. $2601$ , find the strength of the class.

Answer:

Let the number of students $= x$

Each student contributed $x$ Rupees.

Therefore $x^2 = 2601$ or $x = 51$

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Question 6: Find the smallest number by which $588$ be multiplied to get a perfect square number.

Answer:

$588 = 2 \times 2 \times 7 \times 3 \times 7$ . Therefore multiply by $3$ to get a perfect square

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Question 7: Find the smallest number by which $2400$ be multiplied to get a perfect square number. Find the square root of the resulting number.

Answer:

$2400 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5 \times 5$ . Therefore multiply by $6$ . The square root would be $120$

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Question 8: Find the smallest number by which $2592$ be multiplied to get a perfect square number.

Answer:

(i) What is the perfect square number so obtained?

$2592 = 2 \times 2 \times 2 \times 2 \times 2 \times 9 \times 9$ .

Therefore smallest number to be multiplied to $2592$ to get a perfect square is $2$ . Perfect square number $= 5184$

(ii) What is the square root of the resulting number?

Square root of the resulting number is $72$

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Question 9: Find the smallest number by which $1728$ be divided to get a perfect square number.

Answer:

(i) What is the perfect square number so obtained?

$1728 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3$ . Hence divide it by $3$ . The number would be $576$

(ii) Find the square root of this number.

Square root $= 24$

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Question 10: Find the smallest number by which 7776 be divided to get a perfect square number.

(i) What is the resulting number? (ii) What is the square root of the number so obtained?

Answer:

(i) What is the resulting number?

$7776 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 9 \times 9$ . Hence divide this by $6$ . The number would be $1296$

(ii) What is the square root of the number so obtained?

Square root $= 36$

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Question 11: Find the least square number which is exactly divisible by each of the numbers $8, 9, 10$ and $15$ .

Answer:

$8 = 2 \times 2 \times 2$

$9 = 3 \times 3$

$10 = 2 \times 5$

$15 = 3 \times 5$

Therefore the number is $2 \times 2 \times 2 \times 5 \times 3 \times 3 = 360$

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Question 12: Find the square root of each of the following by division method:

i) $961$ ii) $5476$ iii) $11449$ iv) $225625$ v) $4401604$ vi) $9653449$

Answer:

By the long division method, we have

i) $961$

${3 \hspace{0.5cm}} | \overline{9} \hspace{0.5cm} \overline{61} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 31 \\ {\hspace{0.7cm}| } \underline{ 9 \hspace{2.0cm} } \\ 61 {\hspace{0.28cm}| } {\hspace{0.2cm} } \hspace{0.5cm} 61 \\ {\hspace{0.7cm}| } \underline{ {\hspace{0.2cm} } \hspace{0.5cm} 61 } \\ {\hspace{0.7cm}| } {\hspace{0.2cm} } \hspace{0.4cm} 0 \hspace{0.5cm}$

ii) $5476$

${7 \hspace{0.7cm}} | \overline{54} \hspace{0.5cm} \overline{76} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 74 \\ {\hspace{0.9cm}| } \underline{ 49 \hspace{2.0cm} } \\ 144 {\hspace{0.28cm}| } {\hspace{0.2cm} 5} \hspace{0.5cm} 76 \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.2cm} 5} \hspace{0.5cm} 76 } \\ {\hspace{0.9cm}| } {\hspace{0.2cm} } \hspace{0.4cm} 0 \hspace{0.5cm}$

iii) $11449$

${1 \hspace{0.7cm}} | \overline{1} \hspace{0.5cm} \overline{14} \hspace{0.5cm} \overline{49} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 107 \\ {\hspace{0.9cm}| } \underline{ 1 \hspace{3.0cm} } \\ 207 {\hspace{0.28cm}| } {\hspace{0.7cm} 14} \hspace{0.5cm} 49 \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.7cm} 14} \hspace{0.5cm} 49 \hspace{1.5cm} } \\ {\hspace{0.9cm}| } {\hspace{0.7cm} } \hspace{0.7cm} 0 \hspace{0.5cm}$

iv) $225625$

${4 \hspace{0.7cm}} | \overline{22} \hspace{0.5cm} \overline{56} \hspace{0.5cm} \overline{25} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 475 \\ {\hspace{0.9cm}| } \underline{ 16 \hspace{3.0cm} } \\ 87 {\hspace{0.48cm}| } {\hspace{0.3cm} 6} \hspace{0.5cm} 56 \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.3cm} 6} \hspace{0.5cm} 09 \hspace{1.5cm} } \\ 945 {\hspace{0.28cm}| } {\hspace{0.95cm} 47} \hspace{0.5cm} 25 \hspace{0.5cm} \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.95cm} 47} \hspace{0.5cm} 25 \hspace{0.5cm} } \\ {\hspace{0.9cm}| } {\hspace{0.7cm} } \hspace{0.7cm} 0 \hspace{0.5cm}$

v) $4401604$

${4 \hspace{1.0cm}} | \overline{4} \hspace{0.5cm} \overline{40} \hspace{0.5cm} \overline{16} \hspace{0.5cm} \overline{04} \hspace{0.5cm} ( 2098 \\ {\hspace{1.2cm}| } \underline{ 4 \hspace{3.0cm} } \\ 409 {\hspace{0.58cm}| } {\hspace{0.7cm} 40} \hspace{0.5cm} 16 \\ {\hspace{1.2cm}| } \underline{ {\hspace{0.7cm} 36} \hspace{0.5cm} 81 \hspace{1.5cm} } \\ 4188 {\hspace{0.38cm}| } {\hspace{0.7cm} 3} \hspace{0.5cm} 35 \hspace{0.5cm} 04 \\ {\hspace{1.2cm}| } \underline{ {\hspace{0.7cm} 3} \hspace{0.5cm} 35 \hspace{0.5cm} 04 } \\ {\hspace{1.2cm}| } {\hspace{0.7cm} } \hspace{0.7cm} 0 \hspace{0.5cm}$

vi) $9653449$

${3 \hspace{1.0cm}} | \overline{9} \hspace{0.5cm} \overline{65} \hspace{0.5cm} \overline{34} \hspace{0.5cm} \overline{49} \hspace{0.5cm} ( 3107 \\ {\hspace{1.2cm}| } \underline{ 9 \hspace{3.0cm} } \\ 61 {\hspace{0.78cm}| } {\hspace{0.7cm} 65} \hspace{0.5cm} \\ {\hspace{1.2cm}| } \underline{ {\hspace{0.7cm} 61} \hspace{0.5cm} \hspace{1.5cm} } \\ 6207 {\hspace{0.38cm}| } {\hspace{0.7cm} 3} \hspace{0.5cm} 34 \hspace{0.5cm} 49 \\ {\hspace{1.2cm}| } \underline{ {\hspace{0.7cm} 4} \hspace{0.5cm} 34 \hspace{0.5cm} 49 } \\ {\hspace{1.2cm}| } {\hspace{0.7cm} } \hspace{0.7cm} 0 \hspace{0.5cm}$

$\\$

Question 13: The area of a square field is $77841$ sq. meters. Find its perimeter.

Answer:

Area = side $\times$ side $= 77841 = 279$ meter

Perimeter $= 4$ side $= 4 279 = 1116$ sq. meters

$\\$

Question 14: Find the least number which must be subtracted from $7581$ to obtain a perfect square. Find this perfect square and its square root.

Answer:

${8 \hspace{0.5cm}} | \overline{75} \hspace{0.5cm} \overline{81} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 31 \\ {\hspace{0.7cm}| } \underline{ 64 \hspace{2.0cm} } \\ 167 {\hspace{0.08cm}| } {\hspace{0.2cm} 11} \hspace{0.5cm} 81 \\ {\hspace{0.7cm}| } \underline{ {\hspace{0.2cm} 11} \hspace{0.5cm} 69 } \\ {\hspace{0.7cm}| } {\hspace{0.2cm} } \hspace{0.4cm} 12 \hspace{0.5cm}$

Subtract $12$ from $7581$ to obtain a perfect square. The number would be $7569$ and the square root would be $87$ .

$\\$

Question 15: Find the least number which must be subtracted from $43379$ to obtain a perfect Find this perfect square and its square root.

Answer:

${2 \hspace{0.7cm}} | \overline{4} \hspace{0.5cm} \overline{33} \hspace{0.5cm} \overline{79} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 208 \\ {\hspace{0.9cm}| } \underline{ 4 \hspace{3.0cm} } \\ 207 {\hspace{0.28cm}| } {\hspace{0.7cm} 33} \hspace{0.5cm} 79 \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.7cm} 32} \hspace{0.5cm} 64 \hspace{1.5cm} } \\ {\hspace{0.9cm}| } {\hspace{0.7cm} 1} \hspace{0.7cm} 15 \hspace{0.5cm}$

Subtract $115$ from $43379$ to obtain perfect square.

Question 16: Find the least number which must be added to 6203 to obtain a perfect square. Find the perfect square and its square

Answer:

${7 \hspace{0.7cm}} | \overline{62} \hspace{0.5cm} \overline{03} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 78 \\ {\hspace{0.9cm}| } \underline{ 49 \hspace{2.0cm} } \\ 148 {\hspace{0.28cm}| } {\hspace{0.1cm} 13} \hspace{0.5cm} 03 \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.1cm} 11} \hspace{0.5cm} 84 } \\ {\hspace{0.9cm}| } {\hspace{0.2cm} 1 } \hspace{0.4cm} 19 \hspace{0.5cm}$

Therefore $78^2 < 7203 < 79^2$

$79^2 = 6241$

Therefore add $(6241-6203) = 38$ to $6203$ to obtain a perfect square $(6241)$ .

Its square root would be $79$ .

$\\$

Question 17: Find the least number which must be added to 506900 to make it a perfect square. Find this perfect square and its square root.

Answer:

${7 \hspace{0.7cm}} | \overline{50} \hspace{0.5cm} \overline{69} \hspace{0.5cm} \overline{00} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 711 \\ {\hspace{0.9cm}| } \underline{ 49 \hspace{3.0cm} } \\ 141 {\hspace{0.28cm}| } {\hspace{0.3cm} 1} \hspace{0.5cm} 69 \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.3cm} 1} \hspace{0.5cm} 41 \hspace{1.5cm} } \\ 1421 {\hspace{0.08cm}| } {\hspace{0.95cm} 28} \hspace{0.5cm} 00 \hspace{0.5cm} \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.95cm} 14} \hspace{0.5cm} 28 \hspace{0.5cm} }$

Therefore $711^2 < 506900 < 712^2$

$712^2 = 506944$

Therefore add $44$ to $506900$ to make it a perfect square of $712$ .

$\\$

Question 18: Find the greatest number of six digits, which is a perfect square. Find the square root of this number.

Answer:

${9 \hspace{1.0cm}} | \overline{99} \hspace{0.5cm} \overline{99} \hspace{0.5cm} \overline{99} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 999 \\ {\hspace{1.2cm}| } \underline{ 81 \hspace{3.0cm} } \\ 189 {\hspace{0.55cm}| } {\hspace{0.1cm} 18} \hspace{0.5cm} 99 \\ {\hspace{1.2cm}| } \underline{ {\hspace{0.1cm} 17} \hspace{0.5cm} 01 \hspace{1.5cm} } \\ 1989 {\hspace{0.38cm}| } {\hspace{0.1cm} 1} \hspace{0.5cm} 98 \hspace{0.5cm} 99 \\ {\hspace{1.2cm}| } \underline{ {\hspace{0.1cm} 1} \hspace{0.5cm} 79 \hspace{0.5cm} 01 } \\ {\hspace{1.2cm}| } {\hspace{0.7cm} 19} \hspace{0.7cm} 98 \hspace{0.5cm}$

Subtract $1998$ from $999999$ to make a perfect square. The number is $998001$ .

$\\$

Question 19: Find the least number of four digits which is a perfect square.

Answer:

${3 \hspace{0.7cm}} | \overline{10} \hspace{0.5cm} \overline{00} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 31 \\ {\hspace{0.9cm}| } \underline{ 9 \hspace{2.0cm} } \\ 61 {\hspace{0.48cm}| } {\hspace{0.1cm} 1} \hspace{0.5cm} 00 \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.1cm} } \hspace{0.7cm} 61 } \\ {\hspace{0.9cm}| } {\hspace{0.2cm} } \hspace{0.5cm} 39 \hspace{0.5cm}$

Therefore $31^2 < 1000 < 32^2$

$32^2 = 1024$

Therefore add $24$ to $1000$ to get the least number of four digits which is a perfect square which is $1024$ .

$\\$

Question 20: Find the least number by which $69192$ must be (i) decreased (ii) increased (iii) multiplied (iv) divided to make it a perfect.

Answer:

${2 \hspace{0.7cm}} | \overline{6} \hspace{0.5cm} \overline{91} \hspace{0.5cm} \overline{92} \hspace{0.5cm} \overline{} \hspace{0.5cm} ( 263 \\ {\hspace{0.9cm}| } \underline{ 4 \hspace{3.0cm} } \\ 46 {\hspace{0.48cm}| } {\hspace{0.3cm} 2} \hspace{0.5cm} 91 \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.3cm} 2} \hspace{0.5cm} 76 \hspace{1.5cm} } \\ 52 {\hspace{0.48cm}| } {\hspace{0.95cm} 15} \hspace{0.5cm} 92 \hspace{0.5cm} \\ {\hspace{0.9cm}| } \underline{ {\hspace{0.95cm} 15} \hspace{0.5cm} 69 \hspace{0.5cm} } \\ {\hspace{0.9cm}| } {\hspace{0.7cm} } \hspace{0.7cm} 23 \hspace{0.5cm}$

Subtract $23$ from $69192$ to make it a perfect square.

$263^2 < 69191 < 2642, 264^2 = 69696$ . Therefore add $504$ to $69192$ to make it a perfect square.

$69192 = 2 \times 2 \times 2 \times 3 \times 3 \times 31 \times 31$ . Therefore multiply by $2$ to make it a perfect square

Or divide it by $2$ to make it a perfect square.

3 thoughts on “Class 8: Square Roots & Cube Roots – Exercise 7A”

Chaitanya Jha says:

November 2, 2018 at 8:43 pm

The exercises of this chapter is outdated and aren’t the one one which we are studying, because other chapters are up to date and current.

Request you to update this chapter as per the current syllabus, as my exams are near and I want to study badly 😀

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1. ICSE Mathematics Student-Assistant says:
  
  November 5, 2018 at 12:04 pm
  
  concepts should be the same…
  
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Chaitanya Jha says:

November 21, 2018 at 11:15 am

But, I want the same sums, please request you to work on it.

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Class 8: Square Roots & Cube Roots – Exercise 7A

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