3 Mark Question - Maths STD 8 Questions - Vidyadip

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3 Mark Question

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Question 13 Marks

Find the greatest number of three digits which is a perfect square.

Answer

The greatest number with three digits is 999. To find the greatest perfect square with three digits, we must find the smallest number that must be subtracted from 999 in order to get a perfect square. For that, we have to find the square root by the long division method as shown below,

So, 38 must be subtracted from 999 to get a perfect square.
$999-38=961$
$961=31^2$
Hence, the greatest perfect square with three digits is 961 .

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Question 23 Marks

The cost of levelling and turfing a square lawn at Rs. 2.50 per $m ^2$ is Rs 13322.50 . Find the cost of fencing it at Rs. 5 per metre.

Answer

First, we have to find the area of the square lawn, which the total cost divided by the cost of levelling and turfing per square metre,
Area of a square $=\frac{13322.5}{2.5}=5329\text{m}^2$
The length of one side of the square is equal to the square root of the area. we will use the long division method to find it as shown below,

$\therefore$ Length of one side of the square $=73 m$
The circumference of the square is $73 \times 4=292 m$
$\therefore$ Total cost of fencing the lawn at Rs. 5 per metre $=292 \times 5=$ Rs. 1460

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Question 33 Marks

The area of a square field is $80\frac{244}{729}$ square metres. Find the length of each side of the field.

Answer

The length of one side of the square root of the area of the field. Hence, we need to calculate the value of $\sqrt{80\frac{244}{729}}$
We have,
$\sqrt{80\frac{244}{729}}=\sqrt{\frac{58564}{729}}=\frac{\sqrt{58564}}{\sqrt{729}}$
Now, to calculate the square root of the numerator and the denominator,

We know that,
$\sqrt{729}=27$
Therefore, length of one side of the field $=\frac{242}{27}=8\frac{26}{27}\text{m}$

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Question 43 Marks

Find the square root of the following by prime factorization.8281

Answer

Resolving 8281 into prime factors,8281 = 7 × 7 × 13 × 13
$\begin{array}{c|c}7& 8281 \\ \hline 7 & 1183 \\\hline 13&169 \\\hline13&13 \\\hline&1\end{array}$
Grouping the factors into pairs of equal factors,
8281 = (7 × 7) × (13 × 13)
Taking one factor for each pair, we get the square root of 705
7 × 13 = 91

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Question 53 Marks

Find the square root of the following correct to three places of decimal:
7

Answer

We can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.

Hence, the square root of 7 up to three decimal places is 2.646.

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Question 63 Marks

Find the square root of the following by prime factorization.47089

Answer

Resolving 47089 into prime factors,
47089 = 7 × 7 × 31 × 31
$\begin{array}{c|c}7& 47089 \\ \hline 7 & 6727 \\\hline 31&961 \\\hline31&31 \\\hline&1\end{array}$
Grouping the factors into pairs of equal factors,
47089 = (7 × 7) × (31 × 31)
Taking one factor for each pair, we get the square root of 47089
7 × 31 = 217

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Question 73 Marks

Find the smallest number by which 147 must be multiplied so that it becomes a perfect square. Also, find the square root of the number so obtained.

Answer

The prime factorisation of 147
147 = 3 × 7 × 7
Grouping the factors into pairs of equal factors, we get,
147 = 3 × (7 × 7)
The factor, 3 does not have a pair. Therefore, we must multiply 147 by 3 to make a perfect square. The new number is,
(3 × 3) × (7 × 7) = 441
Taking one factor from each pair on the L.H.S, the square root of the new number is 3 × 7, which is equal to 21.

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Question 83 Marks

Find the square root of the following correct to three places of decimal:
2.5

Answer

We can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.

Hence, the square root of 2.5 up to three decimal places is 1.581

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Question 93 Marks

Find the greatest number of 4 digits which is a perfect square.

Answer

The greatest number with four digits is 9999. To find the greatest perfect square with four digits, we must find the smallest number that must be subtracted from 9999 in order to make a perfect square. For that, we have to find the square root of 9999 by the long division method as shown below:

We must subtract 198 from 9999 to make a perfect square,
9999 - 198 = 9801
Hence, the greatest perfect square with four digits is 9801.

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Question 103 Marks

Find the square root of the following correct to three places of decimal:
$287\frac{5}{8}$

Answer

We can find the square root up to four decimal places by expanding $287\frac{5}{8}$ into decimal form up to eight digits to the right of the decimal point as shown below,
$287\frac{5}{8}=287.62500000$
Hence, we have

So, the square root of $287\frac{5}{8}$ up to three decimal places is 16.960

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Question 113 Marks

Find the square root of the following correct to three places of decimal:0.016

Answer

We can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.

Hence, the square root of 0.016 up to three decimal places is 0.126

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Question 123 Marks

Find the square root of the following by prime factorization.
196

Answer

Resolving 196 into prime factors,
196 = 2 × 2 × 7 × 7
$\begin{array}{c|c}2& 196 \\ \hline 2 & 98 \\\hline 7&49 \\\hline 7&7 \\\hline&1\end{array}$
Grouping the factors into pairs of equal factors,
196 = (2 × 2) × (7 × 7)
Taking one factor for each pair, we get the square root of 196
2 × 7 = 14

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Question 133 Marks

Find the least number of 4 digits which is a perfect square.

Answer

The least number with four digits is 1000. To find the least square number with four digits, we must find the smallest number that must be added to 1000 in order to make a perfect square. For that, we have to find the square root of 1000 by the long division method as shown below:

1000 is $24(124-100)$ less than the nearest square number $32^2$. Thus, 24 must be added to 1000 to be a perfect square.
$1000+24=1024$
Hence, the smallest perfect square number with four digits is 1024.

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Question 143 Marks

Find the square root of the following by prime factorization.3013696

Answer

Resolving 3013696 into prime factors,
3013696 = 2 × 2 × 2 × 2 × 2 × 2 × 7 × 7 × 31 × 31
$\begin{array}{c|c}2&3013696 \\ \hline 2 & 1506848 \\\hline 2&753424\\\hline2&376712 \\\hline2&188356\\\hline2&941781\\\hline7&47089\\\hline7&6727\\\hline31&961\\\hline31&31\\\hline&1\end{array}$
Grouping the factors into pairs of equal factors,
3013696 = (2 × 2) × (2 × 2) × (2 × 2) × (7 × 7) × (31 × 31)
Taking one factor for each pair, we get the square root of 3013696
2 × 2 × 2 × 7 × 31 = 1736

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Question 153 Marks

Find the square root of the following correct to three places of decimal:
23.1

Answer

We can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.

Hence, the square root of 23.1 up to three decimal places is 4.806.

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Question 163 Marks

Find the square root of the following correct to three places of decimal:
5

Answer

We can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.

Hence, the square root of 5 up to three decimal places is 2.236

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Question 173 Marks

Show that the following numbers is a perfect square. Also, find the number whose square is the given number in each case:
$1156$

Answer

In problem, factorise the number into its prime factors.
$1156=2 \times 2 \times 17 \times 17$
Grouping the factors into pairs of equal factors, we obtain,
$1156=(2 \times 2) \times(17 \times 17)$
No factors are left over. Hence, 1156 is a perfect square. Moreover, by grouping 1156 into equal factors,
$1156=(2 \times 17) \times(2 \times 17)$
$1156=(2 \times 17)^2$
Hence, 1156 is the square of 34 , which is equal to $2 \times 17$

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Question 183 Marks

Write the prime factorization of the following numbers and hence find their square roots.
7744

Answer

The prime factorisation of 7744,
7744 = 2 × 2 × 2 × 2 × 2 × 2 × 11 × 11
Grouping them into pairs of equal factors, we get:
7744 = (2 × 2) × (2 × 2) × (2 × 2) × (11 × 11)
Taking one factor from each pair, we get,
$\sqrt{7744}=2\times2\times2\times2\times11=88$

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Question 193 Marks

Find the square root of the following correct to three places of decimal:
20

Answer

We can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.

Hence, the square root of 20 up to three decimal places is 4.472

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Question 203 Marks

A school collected Rs. 2304 as fees from its students. If each student paid as many paise as there were students in the school, how many students were there in the school?

Answer

Let S be the number of students.
Let $r$ be the money donated by each student.
The total contribution can be expressed by (S)(r) = Rs. 2304
Since each student paid as many paise as the number of students, then $r=S$. Substituting this in the first equation, we get,
$S \times S=2304$
$S^2=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3$
$S^2=(2 \times 2) \times(2 \times 2) \times(2 \times 2) \times(2 \times 2) \times(3 \times 3)$
$S=2 \times 2 \times 2 \times 2 \times 3=48$
So, there are 48 students in total in the school.

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Question 213 Marks

Find the smallest number by which the given number must be divided so that the resulting number is a perfect square:
1800

Answer

Factorise number into its prime factors.
1800 = 2 × 2 × 2 × 3 × 3 × 5 × 5
$\begin{array}{c|c} 2 & 1800 \\ \hline 2 & 900 \\\hline 2&450 \\\hline 3&225 \\\hline 3&75 \\\hline 5 &25 \\\hline 5&5 \\\hline &1 \end{array}$
Grouping the factors into pairs,
1800 = (2 × 2) × (3 × 3) × (5 × 5) × 2
Here, the factor 2 does not occur in pairs. To be a perfect square, all the factors have to be in pairs. Hence, the smallest number by which 1800 must be divided for it to be a perfect square is 2.

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Question 223 Marks

Find the least square number, exactly divisible by one of the numbers:
6, 9, 15 and 20

Answer

The smallest number divisible by 6, 9, 15 and 20 is their L.C.M., which is equal to 60.
Factorising 60 into its prime factors,
60 = 2 × 2 × 3 × 5
Grouping them into pairs of equal factors,
60 = (2 × 2) × 3 × 5
The factors 3 and 5 are not paired. To make 60 a perfect square, we have to multiply it by 3 × 5, i.e by 15.
The perfect square is 60 × 15, which is equal to 900.

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Question 233 Marks

Find the smallest number by which 180 must be multiplied so that it becomes a perfect square. Also, find the square root of the perfect square so obtained.

Answer

The prime factorisation of 180
180 = 2 × 2 × 3 × 3 × 5
Grouping the factors into pairs of equal factors, we get,
180 = (2 × 2) × (3 × 3) × 5
The factor, 5 does not have a pair. Therefore, we must multiply 180 by 5 to make a perfect square. The new number is,
(2 × 2) × (3 × 3) × (5 × 5) = 900
Taking one factor from each pair on the LHS, the square root of the new number is 2 × 3 × 5, which is equal to 30.

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Question 243 Marks

Find the square root of the following correct to three places of decimal:
237.615

Answer

We can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.

Hence, the square root of 237.615 up to three decimal places is 15.415

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Question 253 Marks

Show that the following numbers is a perfect square. Also, find the number whose square is the given number in case:
$14641$

Answer

In problem, factorise the number into its prime factors.
$14641=11 \times 11 \times 11 \times 11$
Grouping the factors into pairs of equal factors, we obtain,
$14641=(11 \times 11) \times(11 \times 11)$
No factors are left over. Hence, 14641 is a perfect square. The above expression is already grouped into equal factors,
$14641=(11 \times 11) \times(11 \times 11)$
$14641=(11 \times 11)^2$
Hence, 14641 is the square of 121 , which is equal to $11 \times 11$

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Question 263 Marks

Find the square root of 12.0068 correct to four decimal places.

Answer

$\therefore\ \sqrt{12.0068}=3.46508$
We can round it off to four decimal places i.e., 3.4651

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Question 273 Marks

Find the squares of the following numbers using column method. Verify the result by finding the square using the usual multiplication,
$96$

Answer

$(96)^2$
Here $a=9, b=6$

$\text{a}^2$	$\text{2ab}$	$\text{b}^2$
$\ \ \ \ \ (9)^2\\ \ =81\\+\ \ {11}$	$2\times9\times6\\=108\\ \ +3$	$\ \ \ (6)^2\\=36$
$92$	$111$

$(96)^2=96 \times 96=9216$
$\therefore(96)^2=9216$

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Question 283 Marks

Find the greatest number of 5 digits which is a perfect square.

Answer

The greatest number with five digits is 99999. To find the greatest square number with five digits, we must find the smallest number that must be subtracted from 99999 in order to make a perfect square. For that, we have to find the square root of 99999 by the long division method as follows:

Hence, we must subtract 143 from 99999 to get a perfect square,
99999 - 143 = 99856

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Question 293 Marks

Find the square root of the following correct to three places of decimal:
66

Answer

We can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.

Hence, the square root of 66 up to three decimal places is 8.124

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Question 303 Marks

Evaluate $\sqrt{50625}$ and hence find the value of $\sqrt{506.25}+\sqrt{5.0625}$

Answer

We have,
$\sqrt{50625}=\sqrt{3\times3\times3\times3\times5\times5\times5\times5}$
$=3\times3\times5\times5=225$
Next, we will calculate $\sqrt{506.25}$ and $\sqrt{5.0625}$
$\sqrt{506.25}=\sqrt{\frac{50625}{100}}=\frac{\sqrt{50625}}{\sqrt{100}}=\frac{225}{10}=22.5$
$\sqrt{5.0625}=\sqrt{\frac{50625}{10000}}=\frac{\sqrt{50625}}{\sqrt{10000}}=\frac{225}{100}=2.25$
$\sqrt{506.25}+\sqrt{5.0625}=22.5+2.25=24.75$

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Question 313 Marks

A General arranges his soldiers in rows to form a perfect square. He finds that in doing so, 60 soldiers are left out. If the total number of soldiers be 8160, find the number of soldiers in each row.

Answer

60 soldiers are left out.
Remainaing soldiers = 8160 - 60 = 8100
The number of soldiers in each row to form a perfect square would be the square root of 8100. We have to find the square root of 8100 by the long division method as shown below,

Hence, the number of soldiers in each row to from a perfect square is 90

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Question 323 Marks

Find the square root of the following by prime factorization.
529

Answer

Resolving 529 into prime factors,
529 = 23 × 23
$\begin{array}{c|c}23& 529 \\ \hline 23 & 23 \\\hline &1\end{array}$
Grouping the factors into pairs of equal factors,
529 = (23 × 23)
Taking one factor for each pair, we get the square root of 529 as 23

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Question 333 Marks

Find the squares of the following numbers using diagonal method:
$348$

Answer

$(348)^2$

$\therefore\ (348)^2=121104$

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Question 343 Marks

Find the smallest number by which the given number must bew multiplied so that the product is a perfect square:
12150

Answer

Factorise number into its prime factors.
12150 = 2 × 3 × 3 × 3 × 3 × 3 × 5 × 5
$\begin{array}{c|c} 2 & 12150 \\ \hline 3 & 6075 \\\hline 3&2025 \\\hline 3&675 \\\hline 3&225 \\\hline 3&75 \\\hline5&25\\\hline5&5\\\hline&1\end{array}$
Grouping 12150 into pairs of equal factors,
12150 = (3 × 3 × 3 × 3) × (5 × 5) × 2 × 3
Here, 2 and 3 do not occur in pairs. To be a perfect square, every prime factor has to be in pairs. Hence. the smallest number by which 12150 must be multiplied is 2 × 3, i.e. by 6.

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Question 353 Marks

Find the square root of the following by prime factorization.
441

Answer

Resolving 441 into prime factors,
441 = 3 × 3 × 7 × 7
$\begin{array}{c|c}3& 441 \\ \hline 3 & 147 \\\hline 7&49 \\\hline 7&7 \\\hline&1\end{array}$
Grouping the factors into pairs of equal factors,
441 = (3 × 3) × (7 × 7)
Taking one factor for each pair, we get the square root of 441
3 × 7 = 21

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Question 363 Marks

Find the smallest number by which the given number must bew multiplied so that the product is a perfect square:
23805

Answer

Factorise number into its prime factors.
23805 = 3 × 3 × 5 × 23 × 23
$\begin{array}{c|c} 3 & 23805 \\ \hline 3 & 7935 \\\hline 5&2645 \\\hline 23&529 \\\hline 23&23 \\\hline &1 \end{array}$
Grouping 23805 into pairs of equal factors,
23805 = (3 × 3) × (23 × 23) × 5
Here, the factor 5 does not occur in pairs. To be a perfect square, every prime factor has to be in pairs. Hence, the smallest number by which 23805 must be multiplied is 5.

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Question 373 Marks

Find the smallest number by which the given number must be divided so that the resulting number is a perfect square:
2904

Answer

Factorise number into its prime factors.
2904 = 2 × 2 × 2 × 3 × 11 × 11
$\begin{array}{c|c} 2 & 2904 \\ \hline 2 & 1452 \\\hline 2&726 \\\hline 3&363 \\\hline 11&121 \\\hline 11 &11 \\\hline &1 \end{array}$
Grouping the factors into pairs,
2904 = (2 × 2) × (11 × 11) × 2 × 3
Here, the factors 2 and 3 do not occur in pairs. To be a perfect square, all the factors have to be in pairs. Hence, the smallest number by which 2904 must be divided for it to be a perfect square is 2 × 3, i.e. 6.

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Question 383 Marks

Find the square root of the following by prime factorization.190969

Answer

Resolving 190969 into prime factors,
190969 = 19 × 19 × 23 × 23
$\begin{array}{c|c}19&190969 \\ \hline 19 & 10051 \\\hline 23&529 \\\hline23&23 \\\hline&1\end{array}$
Grouping the factors into pairs of equal factors,
190969 = (19 × 19) × (23 × 23)
Taking one factor for each pair, we get the square root of 190969
19 × 23 = 437

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Question 393 Marks

Find the squares of the following numbers using the identity $(a-b)^2=a^2-2 a b+b^2$ :
95

Answer

$(a+b)^2=a^2-a b+a b+b^2$
$(95)^2=(90+5)^2$
$=8100+450+450+25$
$=9025$

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Question 403 Marks

Find the squares of the following numbers using diagonal method:
$171$

Answer

$(171)^2$

$\therefore\ (171)^2=29241$

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Question 413 Marks

Find the square root of the following by prime factorization.27225

Answer

Resolving 27225 into prime factors,
27225 = 3 × 3 × 5 × 5 × 11 × 11
$\begin{array}{c|c}3&27225 \\ \hline 3 & 9075 \\\hline 5&3025\\\hline5&605 \\\hline11&121\\\hline11&11\\\hline&1\end{array}$
Grouping the factors into pairs of equal factors,
27225 = (3 × 3) × (5 × 5) × (11 × 11)
Taking one factor for each pair, we get the square root of 27225
3 × 5 × 11 = 165

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Question 423 Marks

Find the smallest number by which the given number must be divided so that the resulting number is a perfect square:
14283

Answer

Factorise number into its prime factors.
14283 = 3 × 3 × 3 × 23 × 23
$\begin{array}{c|c} 3 & 14283 \\ \hline 3 & 4761 \\\hline 3&1587 \\\hline 23&529 \\\hline 23&23 \\\hline &1 \end{array}$
Grouping the factors into pairs,
14283 = (3 × 3) × (23 × 23) × 3
Here, the factor 3 does not occur in pairs. To be a perfect square, all the factors have to be in pairs. Hence, the smallest number by which 14283 must be divided for it to be a perfect square is 3.

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Question 433 Marks

Find the smallest number by which 4851 must be multiplied so that the product becomes a perfect suqare.

Answer

Prime factorisation of 48514851 = 3 × 3 × 7 × 7 × 11
$\begin{array}{c|c} 3& 4851 \\ \hline 3 & 1617 \\\hline 7&539 \\\hline 7 &77\\\hline11&11\\\hline&1 \end{array}$
Grouping them into pairs of equal factors,
4851 = (3 × 3) × (7 × 7) × 11
The factor, 11 is not paired. The smallest number by which 4851 must be multiplied such that the resulting number is a perfact square is 11

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Question 443 Marks

Write the prime factorization of the following numbers and hence find their square roots.
5929

Answer

The prime factorisation of 9604,
5929 = 7 × 7 × 11 × 11
Grouping them into pairs of equal factors, we get:
5929 = (7 × 7) × (11 × 11)
Taking one factor from each pair, we get,
$\sqrt{5929}=7\times11=77$

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Question 453 Marks

Find the squares of the following numbers using the identity $(a-b)^2=a^2-2 a b+b^2$ :
99

Answer

$(a+b)^2=a^2-a b+a b+b^2$
$(99)^2=(90+9)^2$
$=8100+810+810+81$
$=9801$

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Question 463 Marks

Find the smallest number by which the given number must bew multiplied so that the product is a perfect square:
7688

Answer

Factorise number into its prime factors.
7688 = 2 × 2 × 2 × 31 × 31
$\begin{array}{c|c} 2 & 7688 \\ \hline 2 & 3844 \\\hline 2&1922 \\\hline 31&961 \\\hline 31&31 \\\hline &1 \end{array}$
Grouping 7688 into pairs of equal factors,
7688 = (2 × 2) × (31 × 31) × 2
Here, 2 does not occur in pairs. To be a perfect square, every prime factor has to be in pairs. Hence, the smallest number by which 7688 must be multiplied is 2.

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Question 473 Marks

Find the square root of the following correct to three places of decimal:
17

Answer

We can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.

Hence, the square root of 17 up to three decimal places is 4.123

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Question 483 Marks

Find the square root of the following correct to three places of decimal:
1.7

Answer

We can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.

Hence, the square root of 1.7 up to three decimal places is 1.304

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Question 493 Marks

Find the least square number, exactly divisible by one of the numbers:8, 12, 15 and 20

Answer

The smallest number divisible by 8, 12, 15 and 20 is their L.C.M., which is equal to 120.
Factorising 120 into its prime factors,
120 = 2 × 2 × 2 × 3 × 5
Grouping them into pairs of equal factors,
120 = (2 × 2) × 2 × 3 × 5
The factors 2, 3 and 5 are not paired. To make 120 into a perfect square, we have to multiply it by 2 × 3 × 5, i.e. by 30.
The perfect square is 120 × 30, which is equal to 3600.

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Question 503 Marks

Find the length of a side of a sqiare, whose area is equal to the area of a rectangle with sides 240m and 70m.

Answer

The area of the rectangle $=240 m \times 70 m=16800 m^2$
Given that the area of the square is equal to the area of the rectangle.
Hence, the area of the square will also be $16800 m^2$
The length of one side of a square is the square root of its area.
$\therefore \sqrt{16800}=\sqrt{2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5 \times 5 \times 7}$
$=2 \times 2 \times 5 \sqrt{2 \times 3 \times 7}$
$=20 \sqrt{42} m$
$=129.60 m$
Hence, the length of one side of the square is 129.60 m

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Question 513 Marks

The students of class VIII of a school donated Rs. 2401 for PM'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.

Answer

Let $S$ be the number of students.
Let $r$ be the amount in rupees donated by each student. The total donation can be expressed by, $S \times r=R s .2401$ Since the total amount in rupees is equal to the number of students, $r$ is equal to S .
Substituting this in the first equation:
$S \times S=2401$
$S^2=(7 \times 7) \times(7 \times 7)$
$S=7 \times 7=4 S$
So, there are 49 students in the class.

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Question 523 Marks

Show that the following numbers is a perfect square. Also, find the number whose square is the given number in case:
$2025$

Answer

In problem, factorise the number into its prime factors.
$2025=3 \times 3 \times 3 \times 3 \times 5 \times 5$
Grouping the factors into pairs of equal factors, we obtain,
$2025=(3 \times 3) \times(3 \times 3) \times(5 \times 5)$
No factors are left over. Hence, 2025 is a perfect square. Moreover, by grouping 2025 into equal factors:
$2025=(3 \times 3 \times 5) \times(3 \times 3 \times 5)$
$2025=(3 \times 3 \times 5)^2$
Hence, 2025 is the square of 45 , which is equal to $3 \times 3 \times 5$

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Question 533 Marks

Find the square root of the following correct to three places of decimal:
$\frac{5}{12}$

Answer

We can find the square root up to four decimal places by expanding $\frac{5}{12}$ to decimal form up to eight digits to the right of the decimal point as shown below, $\frac{5}{2}=0.41666666$ Hence, we have,

So, the square root of $\frac{5}{12}$ up to three decimal places is 0.645

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Question 543 Marks

Find the square root of the following by prime factorization.1156

Answer

Resolving 1156 into prime factors,
1156 = 2 × 2 × 17 × 17
$\begin{array}{c|c}2& 1156 \\ \hline 2 & 578 \\\hline 17&289 \\\hline17&17 \\\hline&1\end{array}$
Grouping the factors into pairs of equal factors,
1156 = (2 × 2) × (17 × 17)
Taking one factor for each pair, we get the square root of 1764,
2 × 17 = 34

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Question 553 Marks

Find the squares of the following numbers using the identity $(a-b)^2=a^2-2 a b+b^2$ :
$702$

Answer

$(a+b)^2=a^2-a b+a b+b^2$
$(702)^2=(700+2)^2$
$=490000+1400+1400+4$
$=492804$

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Question 563 Marks

Find the squares of the following numbers using column method. Verify the result by finding the square using the usual multiplication,
37

Answer

$(37)^2$
Here, a = 3, b = 7

$\text{a}^2$	$2\text{ab}$	$\text{b}^2$
$\ \ (3)^2\\=9\\+{4}$	$2\times3\times7\\=42\\+\ \ 4$	$\ \ \ (7)^2\\=49$
$13$	$46$

$(37)^2=37 \times 37=1369$
$(37)^2=1369$

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Question 573 Marks

A welfare association collected Rs. 202500 as donation from the residents. If each paid as many rupees as there were residents, find the number of residents.

Answer

Let R be the number of residents.
Let $r$ be the money in rupees donated by each resident.
Total donation $= R \times r =202500$
Since the money received as donation is the same as the number of residents:
$r=R$
Substituting this in the first equation, we get!
$R \times R=202500$
$R^2=202500$
$R^2=(2 \times 2) \times(5 \times 5) \times(5 \times 5) \times(3 \times 3)^2$
$R=2 \times 5 \times 5 \times 3 \times 3=450$
So, the number of residents is 450

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Question 583 Marks

Find the square root of the following correct to three places of decimal:0.019

Answer

We can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.

Hence, the square root of 0.019 up to three decimal places is 0.138

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Question 593 Marks

Find the square root of the following correct to three places of decimal:0.90

Answer

We can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.

Hence, the square root of 0.9 up to three decimal places is 0.949

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Question 603 Marks

Find the squares of the following numbers using column method. Verify the result by finding the square using the usual multiplication,
54

Answer

$(54)^2$

$\text{a}^2$	$\text{2ab}$	$\text{b}^2$
$\ \ \ (5)^2\\=25\\ +\ { 4}$	$2\times5\times4\\=40\\+\ \ 1$	$\ \ \ (4)^2\\=16$
$29$	$41$

$(54)^2=54 \times 54=2961$
$\therefore(54)^2=2961$

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Question 613 Marks

Find the squares of the following numbers using diagonal method:
$273$

Answer

$(273)^2$

$\therefore\ (273)^2=74529$

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Question 623 Marks

Find the squares of the following numbers using diagonal method:
$295$

Answer

$(295)^2$

$\therefore\ (295)^2=87025$

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Question 633 Marks

Find the square root of 11 correct to five decimal places.

Answer

Using the long division method,

$\therefore\ \sqrt{11}=3.31662$

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Question 643 Marks

Using square root table, find the square root:
4192

Answer

$\sqrt{4192}=\sqrt{2\times2\times2\times2\times2\times131}$
$=2\times2\sqrt{2}\times\sqrt{131}$
The square root of 131 is not listed in the table. Hence, we have to apply long division to find it.

Substituting the values,
$=2\times2\times11.4455$ $\big($Using the table to find $\sqrt{2}\big)$
$=64.75$

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Question 653 Marks

Find the squares of the following numbers using column method. Verify the result by finding the square using the usual multiplication,
$25$

Answer

$(25)^2$
Here, $a = 2, b = 5$

$\text{a}^2$	$2\text{ab}$	$\text{b}^2$
$\ \ (2)^2\\=4\\+{2}\\ \ \ \ \ \overline{6}$	$2\times2\times5\\=20\\+\ \ 2\\ \ \ \ \ \overline{22}$	$\ \ \ (5)^2\\=25$

$(25)^2=25 \times 25=625$
$(25)^2=625$

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Question 663 Marks

Show that the following numbers is a perfect square. Also, find the number whose square is the given number in case:4761

Answer

In problem, factorise the number into its prime factors.
$4761=3 \times 3 \times 23 \times 23$
Grouping the factors into pairs of equal factors, we obtain,
$4761=(3 \times 3) \times(23 \times 23)$
No factors are left over. Hence, 4761 is a perfect square. The above expression is already grouped into equal factors,
$4761=(3 \times 23) \times(3 \times 23)$
$4761=(3 \times 23)^2$
Hence, 4761 is the square of 69 , which is equal to $3 \times 23$.

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Question 673 Marks

Write five numbers for which you cannot decide whether they are squares.

Answer

A number whose unit digit is 2, 3, 7 or 8 cannot be a perfect square. On the other hand, a number whose unit digit is 1, 4, 5, 6, 9 or 0 might be a perfect square (although we will have to verify whether it is a perfect square or not). Applying the above two conditions, we cannot quickly decide whether the following numbers are squares of any numbers,
1111, 1444, 1555, 1666, 1999

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Question 683 Marks

Write the prime factorization of the following numbers and hence find their square roots.
7056

Answer

The prime factorisation of 9604,
7056 = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7
Grouping them into pairs of equal factors, we get:
7056 = (2 × 2) × (2 × 2) × (3 × 3) × (7 × 7)
Taking one factor from each pair, we get,
$\sqrt{7056}=2\times2\times3\times7=84$

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Question 693 Marks

Find the square root of the following correct to three places of decimal:
$\frac{7}{8}$

Answer

We can find the square root up to four decimal places by expanding $\frac{7}{8}$ to decimal form up to eight digits to the right of the decimal point as shown below,
$\frac{7}{8}=0.875$
Hence, we have,

So, the square root of $\frac{7}{8}$ up to three decimal places is 0.935

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Question 703 Marks

Find the square root of the following correct to three places of decimal:0.00064

Answer

We can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.

Hence, the square root of 0.00064 up to three decimal places is 0.025

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Question 713 Marks

Find the squares of the following numbers using diagonal method:
98

Answer

$(98)^2$

$\therefore\ (98)^2=9604$

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Question 723 Marks

Write five numbers which you cannot decide whether they are square just by looking at the unit's digit.

Answer

A number whose unit digit is 2, 3, 7 or 8 cannot be a perfect square. On the other hand, a number whose unit digit is 1, 4, 5, 6, 9 or 0 might be a perfect square although we have to verify that. Applying these two conditions, we cannot determine whether the following numbers are squares just by looking at their unit digits,
1111, 1001, 1555, 1666 and 1999

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Question 733 Marks

Write the prime factorization of the following numbers and hence find their square roots.
9604

Answer

The prime factorisation of 9604,
9604 = 2 × 2 × 7 × 7 × 7 × 7
Grouping them into pairs of equal factors, we get:
9604 = (2 × 2) × (7 × 7) × (7 × 7)
Taking one factor from each pair, we get,
$\sqrt{9604}=2\times7\times7=98$

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Question 743 Marks

Find the square root of the following by prime factorization.7056

Answer

Resolving 7056 into prime factors,
7056 = 2 × 2 × 2 × 2 × 3 × 3 × 7 × 7
$\begin{array}{c|c}2& 7056 \\ \hline 2 & 3528 \\\hline 2&1764 \\\hline2&882\\\hline3&441\\\hline3&147\\\hline7&49\\\hline7&7\\\hline&1\end{array}$
Grouping the factors into pairs of equal factors,
7056 = (2 × 2) × (2 × 2) × (3 × 3) × (7 × 7)
Taking one factor for each pair, we get the square root of 705
2 × 2 × 3 × 7 = 84

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Question 753 Marks

Find the squares of the following numbers using column method. Verify the result by finding the square using the usual multiplication,
$71$

Answer

$(71)^2$
Here, a = 7, b = 1

$\text{a}^2$	$2\text{ab}$	$\text{b}^2$
$\ \ \ (7)^2\\=49\\+\ \ {1}$	$2\times7\times1\\=14$	$\ \ \ (1)^2\\=\ 1$
$50$

$(71)^2=71 \times 71=5041$
$\therefore(71)^2=5041$

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Question 763 Marks

Find the squares of the following numbers using the identity $(a-b)^2=a^2-2 a b+b^2$ :
505

Answer

$(a+b)^2=a^2-a b+a b+b^2$
$(505)^2=(500+5)^2$
$=250000+2500+2500+25$
$=255025$

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Question 773 Marks

Using square root table, find the square root:
21.97

Answer

We have to find $\sqrt{21.97}$
From the square root table, we have,
$\sqrt{21}=\sqrt{3}\times\sqrt{7}=4.583$ and $\sqrt{22}=\sqrt{2}\times\sqrt{11}=4.690$
Their difference is 0.107
Thus, for the difference of 1 (22 - 21), the difference in the values of the square roots is 0.107
For the difference of 0.97, the difference in the values of the values of their square roots is,
0.107 × 0.97 = 0.104
$\therefore\sqrt{21.97}=4.583+0.104\approx4.687$

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Question 783 Marks

Find the square root of the following by prime factorization.1764

Answer

Resolving 1764 into prime factors,
1764 = 2 × 2 × 3 × 3 × 7 × 7
$\begin{array}{c|c}2& 1764 \\ \hline 2 & 882 \\\hline 3&441 \\\hline3&147 \\\hline7&49\\\hline7&7\\\hline&1 \end{array}$
Grouping the factors into pairs of equal factors,
1764 = (2 × 2) × (3 × 3) × (7 × 7)
Taking one factor for each pair, we get the square root of 1764,
2 × 3 × 7 = 42

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Question 793 Marks

Find the square root of the following correct to three places of decimal:0.1

Answer

We can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.

Hence, the square root of 0.1 up to three decimal places is 0.316

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Question 803 Marks

Using square root table, find the square root:
13.21

Answer

From the square root table, we have,
$\sqrt{13}=3.606$ and $\sqrt{14}=\sqrt{2}\times\sqrt{7}=3.742$
Their difference is 0.136
Thus, for the difference of 1 (14 - 13), the difference in the values of the square roots is 0.136
For the difference of 0.21, the difference in the values of their square roots is,
0.136 × 0.21 = 0.02856
$\therefore\sqrt{13.21}=3.606+0.02856\approx3.635$

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Question 813 Marks

Find the square root of the following by prime factorization.586756

Answer

Resolving 586756 into prime factors,
586756 = 2 × 2 × 383 × 383
$\begin{array}{c|c}2&586756 \\ \hline 2 & 293378 \\\hline 383&146689 \\\hline383&383 \\\hline&1\end{array}$
Grouping the factors into pairs of equal factors,
586756 = (2 × 2) × (383 × 383)
Taking one factor for each pair, we get the square root of 586756
2 × 383 = 766

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Question 823 Marks

Find the square root of the following by prime factorization.4096

Answer

Resolving 4096 into prime factors,
4096 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2
$\begin{array}{c|c}2& 4096 \\ \hline 2 & 2048 \\\hline 2&1024 \\\hline2&512 \\\hline2&256\\\hline2&128\\\hline2&64\\\hline2&32\\\hline2&16\\\hline2&8\\\hline2&4\\\hline2&2\\\hline&1\end{array}$
Grouping the factors into pairs of equal factors,
4096 = (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2)
Taking one factor for each pair, we get the square root of 4096,
(2 × 2) × (2 × 2) × (2 × 2) = 64

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Question 833 Marks

Find the squares of the following numbers using the identity $(a-b)^2=a^2-2 a b+b^2$ :
52

Answer

$(a+b)^2=a^2-a b+a b+b^2$
$(52)^2=(50+2)^2$
$=2500+100+100+4$
$=2704$

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Question 843 Marks

Find the smallest number by which 3645 must be divided so that it becomes a perfect square. Also, find the square root of the resulting number.

Answer

The prime factorisation of 3645,
3645 = 3 × 3 × 3 × 3 × 3 × 3 × 5
Grouping the factors into pairs of equal factors, we get,
3645 = (3 × 3) × (3 × 3) × (3 × 3) × 5
The factor, 5 does not have a pair. Therefore, we must divide 3645 by 5 to make a perfect square. The new number is,
(3 × 3) × (3 × 3) × (3 × 3) = 729
Taking one factor from each pair on the L.H.S, the square root of the new number is 3 × 3 × 3, which is equal to 27.

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Question 853 Marks

Find the square root of the following by prime factorization.11664

Answer

Resolving 11664 into prime factors,11664 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 × 3 × 3
$\begin{array}{c|c}2& 11644 \\ \hline 2 & 5832 \\\hline 2&2916 \\\hline2&1458 \\\hline3&729\\\hline3&243\\\hline3&81\\\hline3&27\\\hline3&9\\\hline3&3\\\hline&1\end{array}$
Grouping the factors into pairs of equal factors,
11664 = (2 × 2) × (2 × 2) × (3 × 3) × (3 × 3) × (3 × 3)
Taking one factor for each pair, we get the square root of 11664
2 × 2 × 3 × 3 × 3 = 108

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Question 863 Marks

Find the smallest number by which 1152 must be divided so that it becomes a perfect square. Also, find the square root of the number so obtained.

Answer

The prime factorisation of 1152,
1152 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3
Grouping the factors into pairs of equal factors, we get,
1152 = (2 × 2) × (2 × 2) × (2 × 2) × (3 × 3) × 2
The factor, 2, at the end, does not have a pair. Therefore, we must divide 1152 by 2 to make a perfect square. The new number is,
(2 × 2) × (2 × 2) × (2 × 2) × (3 × 3) = 576
Taking one factor from each pair on the LHS, the square root of the new number is 2 × 2 × 2 × 3, which is equal to 24

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Question 873 Marks

Find the square root of the following correct to three places of decimal:15.3215

Answer

We can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.

Hence, the square root of 15.3215 up to three decimal places is 3.914

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Question 883 Marks

Find the smallest number by which 28812 must be divided so that the quotient becomes a perfect square.

Answer

Prime factorisation of 28812
28812 = 2 × 2 × 3 × 7 × 7 × 7 × 7
$\begin{array}{c|c} 2& 28812 \\ \hline 2 & 14406 \\\hline 3&7203 \\\hline 7 &2401\\\hline7&343\\\hline7&49\\\hline7&7\\\hline&1 \end{array}$
Grouping them into pairs of equal factors,
28812 = (2 × 2) × (7 × 7) × (7 × 7) × 3
The factor, 3 is not paired. The smallest number by which 28812 must be multiplied such that the resulting number is a perfact square is 3

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Question 893 Marks

Simplify:
$\frac{\sqrt{0.2304}\ +\sqrt{0.1764}}{\sqrt{0.2304}\ -\sqrt{0.1764}}$

Answer

We have,
$\sqrt{0.2304}=\sqrt{\frac{2304}{10000}}$
$=\frac{\sqrt{2\times2\times2\times2\times2\times2\times3\times3}}{\sqrt{10000}}$
$\frac{2\times2\times2\times2\times3}{100}$
$=0.48$
$\sqrt{0.1764}=\sqrt{\frac{1764}{10000}}$
$=\frac{\sqrt{2\times2\times3\times3\times7\times7}}{\sqrt{10000}}$
$=\frac{2\times3\times7}{100}$
$=0.42$
$\frac{\sqrt{0.2304}\ +\sqrt{0.1764}}{\sqrt{0.2304}\ -\sqrt{0.1764}}=\frac{0.48+0.42}{0.48-0.42}$
$=\frac{0.9}{0.06}$
$=15$

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Question 903 Marks

Find the square root of the following by prime factorization.24336

Answer

Resolving 24336 into prime factors,
24336 = 2 × 2 × 2 × 2 × 3 × 3 × 13 × 13
$\begin{array}{c|c}2& 24336 \\ \hline 2 & 12168 \\\hline 2&6084 \\\hline2&3042 \\\hline3&1521 \\\hline 3&507\\\hline13&169\\\hline13&13\\\hline&1\end{array}$
Grouping the factors into pairs of equal factors,
24336 = (2 × 2) × (2 × 2) × (3 × 3) × (13 × 13)
Taking one factor for each pair, we get the square root of 24336
2 × 2 × 3 × 13 = 156

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Question 913 Marks

Find the square root of the following correct to three places of decimal:
427

Answer

We can find the square root up to three decimal places by using long division until we get four decimal places and then rounding it to three decimal places.

Hence, the square root of 427 up to three decimal places is 20.664

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Question 923 Marks

A PT teacher wants to arrange maximum possible number of 6000 students in a field such that the number of rows is equal to the number of columns. Find the number of rows if 71 were left out after arrangement.

Answer

Since 71 students were left out, there are only 5929 (6000 - 71) students remaining.
Hence, the number of rows or columns is simply the square root of 5929.
Factorising 5929 into its prime factors,
5929 = 7 × 7 × 11 × 11
Grouping them into pairs of equal factors,
5929 = (7 × 7) × (11 × 11)
The square root of 5929
$=\sqrt{5929}=7\times11=77$
Hence, in the arrangement, there were 77 rows of students.

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