Question 13 Marks
Find the greatest number of three digits which is a perfect square.
AnswerThe 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 $961.$ View full question & answer→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.
AnswerFirst, 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 $= 73m$ The circumference of the square is $73 \times 4 = 292m$
$\therefore$ Total cost of fencing the lawn at Rs. $5$ per metre $= 292 \times 5 = Rs. 1460$ View full question & answer→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.
AnswerThe 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 $=\frac{242}{27}=8\frac{26}{27}\text{m}$ View full question & answer→Question 43 Marks
Find the square root of the following by prime factorization.8281
AnswerResolving $8281$ into prime factors,$8281 = 7 \times 7 \times 13 \times 13$
$\begin{array}{c|c}7& 8281 \\ \hline 7 & 1183 \\\hline 13&169 \\\hline13&13 \\\hline&1\end{array}$
Grouping the factors into pairs of equal factors,
$8281 = (7 \times 7) \times (13 \times 13)$
Taking one factor for each pair, we get the square root of $705$
$7 \times 13 = 91$
View full question & answer→Question 53 Marks
Find the square root of the following correct to three places of decimal:
$7$
AnswerWe 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 $2.646.$ View full question & answer→Question 63 Marks
Find the square root of the following by prime factorization.$47089$
AnswerResolving $47089$ into prime factors, $47089 = 7 \times 7 \times 31 \times 31$
$\begin{array}{c|c}7& 47089 \\ \hline 7 & 6727 \\\hline 31&961 \\\hline31&31 \\\hline&1\end{array}$
Grouping the factors into pairs of equal factors, $47089 = (7 \times 7) \times (31 \times 31)$
Taking one factor for each pair, we get the square root of $47089 7 \times 31 = 217$
View full question & answer→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.
AnswerThe 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.
View full question & answer→Question 83 Marks
Find the square root of the following correct to three places of decimal:
$2.5$
AnswerWe 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 $1.581$ View full question & answer→Question 93 Marks
Find the greatest number of $4$ digits which is a perfect square.
AnswerThe 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 $9801.$ View full question & answer→Question 103 Marks
Find the square root of the following correct to three places of decimal: $287\frac{5}{8}$
AnswerWe 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 $16.960$ View full question & answer→Question 113 Marks
Find the square root of the following correct to three places of decimal:$0.016$
AnswerWe 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 $0.126$ View full question & answer→Question 123 Marks
Find the square root of the following by prime factorization. $196$
AnswerResolving 196 into prime factors, $196 = 2 \times 2 \times 7 \times 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, $196 = (2 \times 2) \times (7 \times 7$)
Taking one factor for each pair, we get the square root of $196 2 \times 7 = 14$
View full question & answer→Question 133 Marks
Find the least number of $4$ digits which is a perfect square.
AnswerThe 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 $1024.$ View full question & answer→Question 143 Marks
Find the square root of the following by prime factorization.$3013696$
AnswerResolving $3013696$ into prime factors, $3013696 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 7 \times 7 \times 31 \times 31$
$\begin{array}{c|c}2&3013696 \\ \hline 2 & 1506848 \\\hline 2&753424\\\hline2&376712 \\\hline2&188356\\\hline2&941781\\\hline7&47089\\\hline7&6727\\\hline31&961\\\hline31&31\\\hline&1\end{array}$
Grouping the factors into pairs of equal factors, $3013696 = (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (7 \times 7) \times (31 \times 31)$ Taking one factor for each pair, we get the square root of $3013696 2 \times 2 \times 2 \times 7 \times 31 = 1736$
View full question & answer→Question 153 Marks
Find the square root of the following correct to three places of decimal: $23.1$
AnswerWe 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 $4.806.$ View full question & answer→Question 163 Marks
Find the square root of the following correct to three places of decimal: $5$
AnswerWe 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 $2.236$ View full question & answer→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: $1156$
AnswerIn 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 $2 \times 17$
View full question & answer→Question 183 Marks
Write the prime factorization of the following numbers and hence find their square roots. $7744$
AnswerThe prime factorisation of $7744, 7744 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 11 \times 11$
Grouping them into pairs of equal factors, we get: $7744 = (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (11 \times 11)$
Taking one factor from each pair, we get, $\sqrt{7744}=2\times2\times2\times2\times11=88$
View full question & answer→Question 193 Marks
Find the square root of the following correct to three places of decimal: $20$
AnswerWe 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 $4.472$ View full question & answer→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?
AnswerLet 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.
View full question & answer→Question 213 Marks
Find the smallest number by which the given number must be divided so that the resulting number is a perfect square: $1800$
AnswerFactorise number into its prime factors. $1800 = 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5$
$\begin{array}{c|c} 2 & 1800 \\ \hline 2 & 900 \\\hline 2&450 \\\hline 3&225 \\\hline 3&75 \\\hline 5 &25 \\\hline 5&5 \\\hline &1 \end{array}$
Grouping the factors into pairs, $1800 = (2 \times 2) \times (3 \times 3) \times (5 \times 5) \times 2$
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 $2.$
View full question & answer→Question 223 Marks
Find the least square number, exactly divisible by one of the numbers:
$6, 9, 15$ and $20$
Answersolution 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,
$60=2 \times 2 \times 3 \times 5$
Grouping them into pairs of equal factors,
$60=(2 \times 2) \times 3 \times 5$
The factors $3$ and $5$ are not paired. To make $60$ a perfect square, we have to multiply it by $3 \times 5$, i.e by 15 . The perfect square is $60 \times 15$, which is equal to $900 .$
View full question & answer→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.
AnswerThe prime factorisation of $180$
$180 = 2 \times 2 \times 3 \times 3 \times 5$
Grouping the factors into pairs of equal factors, we get,
$180 = (2 \times 2) \times (3 \times 3) \times 5$
The factor, 5 does not have a pair. Therefore, we must multiply $180$ by $5$ to make a perfect square. The new number is,
$(2 \times 2) \times (3 \times 3) \times (5 \times 5) = 900$
Taking one factor from each pair on the $LHS$, the square root of the new number is $2 \times 3 \times 5$, which is equal to $30.$
View full question & answer→Question 243 Marks
Find the square root of the following correct to three places of decimal: $237.615$
AnswerWe 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 $15.415$ View full question & answer→Question 253 Marks
Show that the following numbers is a perfect square. Also, find the number whose square is the given number in case:$14641$
AnswerIn problem, factorise the number into its prime factors.
$14641 = 11 × 11 × 11 × 11$
Grouping the factors into pairs of equal factors, we obtain,
$14641 = (11 × 11) × (11 × 11)$
No factors are left over.Hence, 14641 is a perfect square. The above expression is already grouped into equal factors,
$14641 = (11 × 11) × (11 × 11)$
$14641 = (11 × 11)^2$
Hence, $14641$ is the square of $121$, which is equal to $11 × 11$
View full question & answer→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., $3.4651$ View full question & answer→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 $a = 9, b = 6$
| $\text{a}^2$ |
$\text{2ab}$ |
$\text{b}^2$ |
| $\ \ \ \ \ (9)^2\\ \ =81\\+\ \ {11}$ |
$2\times9\times6\\=108\\ \ +3$ |
$\ \ \ (6)^2\\=36$ |
| $92$ |
$111$ |
|
$(96)^2= 96 × 96 = 9216$
$\therefore (96)^2= 9216$ View full question & answer→Question 283 Marks
Find the greatest number of $5$ digits which is a perfect square.
AnswerThe 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,
$99999 - 143 = 99856$ View full question & answer→Question 293 Marks
Find the square root of the following correct to three places of decimal: $66$
AnswerWe 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 $8.124$ View full question & answer→Question 303 Marks
Evaluate $\sqrt{50625}$ and hence find the value of $\sqrt{506.25}+\sqrt{5.0625}$
AnswerWe 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 $\sqrt{5.0625}$
$\sqrt{506.25}=\sqrt{\frac{50625}{100}}=\frac{\sqrt{50625}}{\sqrt{100}}=\frac{225}{10}=22.5$
$\sqrt{5.0625}=\sqrt{\frac{50625}{10000}}=\frac{\sqrt{50625}}{\sqrt{10000}}=\frac{225}{100}=2.25$
$\sqrt{506.25}+\sqrt{5.0625}=22.5+2.25=24.75$
View full question & answer→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 $90$ View full question & answer→Question 323 Marks
Find the square root of the following by prime factorization. $529$
AnswerResolving $529$ into prime factors, $529 = 23 \times 23$
$\begin{array}{c|c}23& 529 \\ \hline 23 & 23 \\\hline &1\end{array}$
Grouping the factors into pairs of equal factors, $529 = (23 \times 23)$
Taking one factor for each pair, we get the square root of $529$ as $23$
View full question & answer→Question 333 Marks
Find the squares of the following numbers using diagonal method: $348$
Answer$(348)^2$
$\therefore\ (348)^2=121104$ View full question & answer→Question 343 Marks
Find the smallest number by which the given number must bew multiplied so that the product is a perfect square: $12150$
AnswerFactorise number into its prime factors. $12150 = 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 5 \times 5$
$\begin{array}{c|c} 2 & 12150 \\ \hline 3 & 6075 \\\hline 3&2025 \\\hline 3&675 \\\hline 3&225 \\\hline 3&75 \\\hline5&25\\\hline5&5\\\hline&1\end{array}$
Grouping $12150$ into pairs of equal factors, $12150 = (3 \times 3 \times 3 \times 3) \times (5 \times 5) \times 2 \times 3$
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 $2 \times 3$, i.e. by $6.$
View full question & answer→Question 353 Marks
Find the square root of the following by prime factorization. $441$
AnswerResolving $441$ into prime factors, $441 = 3 \times 3 \times 7 \times 7$
$\begin{array}{c|c}3& 441 \\ \hline 3 & 147 \\\hline 7&49 \\\hline 7&7 \\\hline&1\end{array}$
Grouping the factors into pairs of equal factors, $441 = (3 \times 3) \times (7 \times 7)$
Taking one factor for each pair, we get the square root of $441 3 \times 7 = 21$
View full question & answer→Question 363 Marks
Find the smallest number by which the given number must bew multiplied so that the product is a perfect square: $23805$
AnswerFactorise number into its prime factors. $23805 = 3 \times 3 \times 5 \times 23 \times 23$
$\begin{array}{c|c} 3 & 23805 \\ \hline 3 & 7935 \\\hline 5&2645 \\\hline 23&529 \\\hline 23&23 \\\hline &1 \end{array}$
Grouping 23805 into pairs of equal factors, $23805 = (3 \times 3) \times (23 \times 23) \times 5$
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 $5.$
View full question & answer→Question 373 Marks
Find the smallest number by which the given number must be divided so that the resulting number is a perfect square: $2904$
AnswerFactorise number into its prime factors. $2904 = 2 \times 2 \times 2 \times 3 \times 11 \times 11$
$\begin{array}{c|c} 2 & 2904 \\ \hline 2 & 1452 \\\hline 2&726 \\\hline 3&363 \\\hline 11&121 \\\hline 11 &11 \\\hline &1 \end{array}$
Grouping the factors into pairs, $2904 = (2 \times 2) \times (11 \times 11) \times 2 \times 3$
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 $2 \times 3,$ i.e. 6.
View full question & answer→Question 383 Marks
Find the square root of the following by prime factorization.$190969$
AnswerResolving 190969 into prime factors, $190969 = 19 \times 19 \times 23 \times 23$
$\begin{array}{c|c}19&190969 \\ \hline 19 & 10051 \\\hline 23&529 \\\hline23&23 \\\hline&1\end{array}$
Grouping the factors into pairs of equal factors, $190969 = (19 \times 19) \times (23 \times 23)$
Taking one factor for each pair, we get the square root of $190969 19 \times 23 = 437$
View full question & answer→Question 393 Marks
Find the squares of the following numbers using the identity $(a - b)^2= a^2- 2ab + b^2: 95$
View full question & answer→Question 403 Marks
Find the squares of the following numbers using diagonal method: $171$
Answer$(171)^2$
$\therefore\ (171)^2=29241$ View full question & answer→Question 413 Marks
Find the square root of the following by prime factorization.$27225$
AnswerResolving $27225$ into prime factors, $27225 = 3 \times 3 \times 5 \times 5 \times 11 \times 11$
$\begin{array}{c|c}3&27225 \\ \hline 3 & 9075 \\\hline 5&3025\\\hline5&605 \\\hline11&121\\\hline11&11\\\hline&1\end{array}$
Grouping the factors into pairs of equal factors, $27225 = (3 \times 3) \times (5 \times 5) \times (11 \times 11)$
Taking one factor for each pair, we get the square root of $27225 3 \times 5 \times 11 = 165$
View full question & answer→Question 423 Marks
Find the smallest number by which the given number must be divided so that the resulting number is a perfect square: $14283$
AnswerFactorise number into its prime factors. $14283 = 3 \times 3 \times 3 \times 23 \times 23$
$\begin{array}{c|c} 3 & 14283 \\ \hline 3 & 4761 \\\hline 3&1587 \\\hline 23&529 \\\hline 23&23 \\\hline &1 \end{array}$
Grouping the factors into pairs, $14283 = (3 \times 3) \times (23 \times 23) \times 3$
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 $3.$
View full question & answer→Question 433 Marks
Find the smallest number by which $4851$ must be multiplied so that the product becomes a perfect suqare.
AnswerPrime factorisation of $48514851 = 3 \times 3 \times 7 \times 7 \times 11$
$\begin{array}{c|c} 3& 4851 \\ \hline 3 & 1617 \\\hline 7&539 \\\hline 7 &77\\\hline11&11\\\hline&1 \end{array}$
Grouping them into pairs of equal factors,
$4851 = (3 \times 3) \times (7 \times 7) \times 11$
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 $11$
View full question & answer→Question 443 Marks
Write the prime factorization of the following numbers and hence find their square roots.
$5929$
AnswerThe prime factorisation of $9604,$
$5929 = 7 \times 7 \times 11 \times 11$
Grouping them into pairs of equal factors, we get:
$5929 = (7 \times 7) \times (11 \times 11)$
Taking one factor from each pair, we get,
$\sqrt{5929}=7\times11=77$
View full question & answer→Question 453 Marks
Find the squares of the following numbers using the identity $(a-b)^2=a^2-2 a b+b^2 \cdot 99$
Answer$ (a+b)^2=a^2-a b+a b+b^2 $
$ (99)^2=(90+9)^2$
$ =8100+810+810+81 $
$ =9801 $
View full question & answer→Question 463 Marks
Find the smallest number by which the given number must bew multiplied so that the product is a perfect square: $7688$
AnswerFactorise number into its prime factors. $7688 = 2 \times 2 \times 2 \times 31 \times 31$
$\begin{array}{c|c} 2 & 7688 \\ \hline 2 & 3844 \\\hline 2&1922 \\\hline 31&961 \\\hline 31&31 \\\hline &1 \end{array}$
Grouping $7688$ into pairs of equal factors, $7688 = (2 \times 2) \times (31 \times 31) \times 2$
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 $2.$
View full question & answer→Question 473 Marks
Find the square root of the following correct to three places of decimal:
$17$
AnswerWe 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 $4.123$ View full question & answer→Question 483 Marks
Find the square root of the following correct to three places of decimal:
$1.7$
AnswerWe 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 $1.304$ View full question & answer→Question 493 Marks
Find the least square number, exactly divisible by one of the numbers:$8, 12, 15$ and $20$
AnswerThe 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,
$120 = 2 \times 2 \times 2 \times 3 \times 5$
Grouping them into pairs of equal factors,
$120 = (2 \times 2) \times 2 \times 3 \times 5$
The factors $2, 3$ and $5$ are not paired. To make $120$ into a perfect square, we have to multiply it by $2 \times 3 \times 5$, i.e. by $30.$
The perfect square is $120 \times 30$, which is equal to $3600.$
View full question & answer→Question 503 Marks
Find the length of a side of a sqiare, whose area is equal to the area of a rectangle with sides $240\ m$ and $70\ m.$
AnswerThe area of the rectangle $= 240m × 70m = 16800m^2$
Given that the area of the square is equal to the area of the rectangle.
Hence, the area of the square will also be $16800m^2$
The length of one side of a square is the square root of its area. $\therefore\sqrt{16800}=\sqrt{2\times2\times2\times2\times2\times3\times5\times5\times7}$
$=2\times2\times5\sqrt{2\times3\times7}$
$=20\sqrt{42}\text{m}$
$=129.60\text{m}$
Hence, the length of one side of the square is $129.60\ m$
View full question & answer→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.
AnswerLet $S$ be the number of students.
Let r be the amount in rupees donated by each student. The total donation can be expressed by, $S \times r = Rs. 2401$
Since the total amount in rupees is equal to the number of students, r is equal to $S.$
Substituting this in the first equation:
$S \times S = 2401$
$S^2= (7 \times 7) \times (7 \times 7)$
$S = 7 \times 7 = 49$
So, there are $49$ students in the class.
View full question & answer→Question 523 Marks
Show that the following numbers is a perfect square. Also, find the number whose square is the given number in case: $2025$
AnswerIn 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 $3 \times 3 \times 5$
View full question & answer→Question 533 Marks
Find the square root of the following correct to three places of decimal:
$\frac{5}{12}$
AnswerWe 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 $0.645$ View full question & answer→Question 543 Marks
Find the square root of the following by prime factorization.$1156$
AnswerResolving 1156 into prime factors, $1156 = 2 \times 2 \times 17 \times 17$
$\begin{array}{c|c}2& 1156 \\ \hline 2 & 578 \\\hline 17&289 \\\hline17&17 \\\hline&1\end{array}$
Grouping the factors into pairs of equal factors, $1156 = (2 \times 2) \times (17 \times 17)$
Taking one factor for each pair, we get the square root of $1764, 2 \times 17 = 34$
View full question & answer→Question 553 Marks
Find the squares of the following numbers using the identity $ (a+b)^2=a^2-a b+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 $
View full question & answer→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, $a = 3, b = 7$
|
$\text{a}^2$
|
$2\text{ab}$
|
$\text{b}^2$
|
|
$\ \ (3)^2\\=9\\+{4}$
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$2\times3\times7\\=42\\+\ \ 4$
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$\ \ \ (7)^2\\=49$
|
|
$13$
|
$46$
|
|
$(37)^2= 37 × 37 = 1369$
$ (37)^2= 1369$ View full question & answer→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.
AnswerLet 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 $450$
View full question & answer→Question 583 Marks
Find the square root of the following correct to three places of decimal:$0.019$
AnswerWe 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 $0.138$ View full question & answer→Question 593 Marks
Find the square root of the following correct to three places of decimal:$0.90$
AnswerWe 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 $0.949$ View full question & answer→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$(54)^2$
| $\text{a}^2$ |
$\text{2ab}$ |
$\text{b}^2$ |
| $\ \ \ (5)^2\\=25\\ +\ { 4}$ |
$2\times5\times4\\=40\\+\ \ 1$ |
$\ \ \ (4)^2\\=16$ |
| $29$ |
$41$ |
|
$(54)^2= 54 × 54 = 2961$
$\therefore (54)^2= 2961$ View full question & answer→Question 613 Marks
Find the squares of the following numbers using diagonal method: $273$
Answer$(273)^2$
$\therefore\ (273)^2=74529$ View full question & answer→Question 623 Marks
Find the squares of the following numbers using diagonal method: $295$
Answer$(295)^2$
$\therefore\ (295)^2=87025$ View full question & answer→Question 633 Marks
Find the square root of $11$ correct to five decimal places.
AnswerUsing the long division method,
$\therefore\ \sqrt{11}=3.31662$ View full question & answer→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 $\sqrt{2}\big)$
$=64.75$ View full question & answer→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, $a = 2, b = 5$
| $\text{a}^2$ |
$2\text{ab}$ |
$\text{b}^2$ |
| $\ \ (2)^2\\=4\\+{2}\\ \ \ \ \ \overline{6}$ |
$2\times2\times5\\=20\\+\ \ 2\\ \ \ \ \ \overline{22}$ |
$\ \ \ (5)^2\\=25$ |
$(25)^2= 25 \times 25 = 625$
$(25)^2= 625$ View full question & answer→Question 663 Marks
Show that the following numbers is a perfect square. Also, find the number whose square is the given number in case:$4761$
AnswerIn 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 $3 \times 23.$
View full question & answer→Question 673 Marks
Write five numbers for which you cannot decide whether they are squares.
AnswerA 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, $1111, 1444, 1555, 1666, 1999$
View full question & answer→Question 683 Marks
Write the prime factorization of the following numbers and hence find their square roots. $7056$
AnswerThe prime factorisation of $9604, 7056 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7 \times 7$
Grouping them into pairs of equal factors, we get: $7056 = (2 \times 2) \times (2 \times 2) \times (3 \times 3) \times (7 \times 7)$
Taking one factor from each pair, we get, $\sqrt{7056}=2\times2\times3\times7=84$
View full question & answer→Question 693 Marks
Find the square root of the following correct to three places of decimal: $\frac{7}{8}$
AnswerWe 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 $0.935$ View full question & answer→Question 703 Marks
Find the square root of the following correct to three places of decimal:$0.00064$
AnswerWe 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 $0.025$ View full question & answer→Question 713 Marks
Find the squares of the following numbers using diagonal method: $98$
Answer$(98)^2$
$\therefore\ (98)^2=9604$ View full question & answer→Question 723 Marks
Write five numbers which you cannot decide whether they are square just by looking at the unit's digit.
AnswerA 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 $1999$
View full question & answer→Question 733 Marks
Write the prime factorization of the following numbers and hence find their square roots. $9604$
AnswerThe prime factorisation of $9604, 9604 = 2 \times 2 \times 7 \times 7 \times 7 \times 7$
Grouping them into pairs of equal factors,
we get: $9604 = (2 \times 2) \times (7 \times 7) \times (7 \times 7)$ Taking one factor from each pair,
we get, $\sqrt{9604}=2\times7\times7=98$
View full question & answer→Question 743 Marks
Find the square root of the following by prime factorization.$7056$
AnswerResolving 7056 into prime factors, $7056 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7 \times 7$
$\begin{array}{c|c}2& 7056 \\ \hline 2 & 3528 \\\hline 2&1764 \\\hline2&882\\\hline3&441\\\hline3&147\\\hline7&49\\\hline7&7\\\hline&1\end{array}$
Grouping the factors into pairs of equal factors, $7056 = (2 \times 2) \times (2 \times 2) \times (3 \times 3) \times (7 \times 7)$
Taking one factor for each pair, we get the square root of $705 2 \times 2 \times 3 \times 7 = 84$
View full question & answer→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, $a = 7, b = 1$
| $\text{a}^2$ |
$2\text{ab}$ |
$\text{b}^2$ |
| $\ \ \ (7)^2\\=49\\+\ \ {1}$ |
$2\times7\times1\\=14$ |
$\ \ \ (1)^2\\=\ 1$ |
| $50$ |
|
|
$(71)^2= 71 \times 71 = 5041$
$\therefore (71)^2= 5041$ View full question & answer→Question 763 Marks
Find the squares of the following numbers using the identity $ (a+b)^2=a^2-a b+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 $
View full question & answer→Question 773 Marks
Using square root table, find the square root: $21.97$
Answersolution 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, $0.107 \times 0.97=0.104 \therefore \sqrt{21.97}=4.583+0.104 \approx 4.687$
View full question & answer→Question 783 Marks
Find the square root of the following by prime factorization.$1764$
AnswerResolving $1764$ into prime factors,
$1764 = 2 \times 2 \times 3 \times 3 \times 7 \times 7$
$\begin{array}{c|c}2& 1764 \\ \hline 2 & 882 \\\hline 3&441 \\\hline3&147 \\\hline7&49\\\hline7&7\\\hline&1 \end{array}$
Grouping the factors into pairs of equal factors,
$1764 = (2 \times 2) \times (3 \times 3) \times (7 \times 7)$
Taking one factor for each pair, we get the square root of $1764,$
$2 \times 3 \times 7 = 42$
View full question & answer→Question 793 Marks
Find the square root of the following correct to three places of decimal:$0.1$
AnswerWe 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 $0.316$ View full question & answer→Question 803 Marks
Using square root table, find the square root:
$13.21$
AnswerFrom 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,
$0.136 \times 0.21 = 0.02856$
$\therefore\sqrt{13.21}=3.606+0.02856\approx3.635$
View full question & answer→Question 813 Marks
Find the square root of the following by prime factorization.$586756$
AnswerResolving 586756 into prime factors, $586756 = 2 \times 2 \times 383 \times 383$
$\begin{array}{c|c}2&586756 \\ \hline 2 & 293378 \\\hline 383&146689 \\\hline383&383 \\\hline&1\end{array}$
Grouping the factors into pairs of equal factors, $586756 = (2 \times 2) \times (383 \times 383)$ Taking one factor for each pair, we get the square root of $586756 2 \times 383 = 766$
View full question & answer→Question 823 Marks
Find the square root of the following by prime factorization.$4096$
AnswerResolving 4096 into prime factors, $4096 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$
$\begin{array}{c|c}2& 4096 \\ \hline 2 & 2048 \\\hline 2&1024 \\\hline2&512 \\\hline2&256\\\hline2&128\\\hline2&64\\\hline2&32\\\hline2&16\\\hline2&8\\\hline2&4\\\hline2&2\\\hline&1\end{array}$ Grouping the factors into pairs of equal factors, $4096 = (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (2 \times 2)$
Taking one factor for each pair, we get the square root of $4096, (2 \times 2) \times (2 \times 2) \times (2 \times 2) = 64$
View full question & answer→Question 833 Marks
Find the squares of the following numbers using the identity $(a+b)^2=a^2-a b+a b+b^2: 52$
Answer$(a+b)^2=a^2-a b+a b+b^2$
$(52)^2=(50+2)^2$
$=2500+100+100+4$
$=2704$
View full question & answer→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.
AnswerThe prime factorisation of $3645,$
$3645 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 5$
Grouping the factors into pairs of equal factors, we get,
$3645 = (3 \times 3) \times (3 \times 3) \times (3 \times 3) \times 5$
The factor, 5 does not have a pair. Therefore, we must divide $3645$ by $5$ to make a perfect square. The new number is,
$(3 \times 3) \times (3 \times 3) \times (3 \times 3) = 729$
Taking one factor from each pair on the $L.H.S$, the square root of the new number is $3 \times 3 \times 3$, which is equal to $27.$
View full question & answer→Question 853 Marks
Find the square root of the following by prime factorization.$11664$
AnswerResolving 11664 into prime factors,$11664 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3$
$\begin{array}{c|c}2& 11644 \\ \hline 2 & 5832 \\\hline 2&2916 \\\hline2&1458 \\\hline3&729\\\hline3&243\\\hline3&81\\\hline3&27\\\hline3&9\\\hline3&3\\\hline&1\end{array}$
Grouping the factors into pairs of equal factors,
$11664 = (2 \times 2) \times (2 \times 2) \times (3 \times 3) \times (3 \times 3) \times (3 \times 3)$
Taking one factor for each pair, we get the square root of $11664$
$2 \times 2 \times 3 \times 3 \times 3 = 108$
View full question & answer→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.
AnswerThe prime factorisation of $1152, 1152 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3$
Grouping the factors into pairs of equal factors, we get, $1152 = (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (3 \times 3) \times 2$
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, $(2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (3 \times 3) = 576$
Taking one factor from each pair on the $LHS$, the square root of the new number is $2 \times 2 \times 2 \times 3$, which is equal to $24$
View full question & answer→Question 873 Marks
Find the square root of the following correct to three places of decimal:$15.3215$
AnswerWe 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 $3.914$ View full question & answer→Question 883 Marks
Find the smallest number by which $28812$ must be divided so that the quotient becomes a perfect square.
AnswerPrime factorisation of $28812 28812 = 2 \times 2 \times 3 \times 7 \times 7 \times 7 \times 7$
$\begin{array}{c|c} 2& 28812 \\ \hline 2 & 14406 \\\hline 3&7203 \\\hline 7 &2401\\\hline7&343\\\hline7&49\\\hline7&7\\\hline&1 \end{array}$
Grouping them into pairs of equal factors, $28812 = (2 \times 2) \times (7 \times 7) \times (7 \times 7) \times 3$
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 $3$
View full question & answer→Question 893 Marks
Simplify: $\frac{\sqrt{0.2304}\ +\sqrt{0.1764}}{\sqrt{0.2304}\ -\sqrt{0.1764}}$
AnswerWe have, $\sqrt{0.2304}=\sqrt{\frac{2304}{10000}}$
$=\frac{\sqrt{2\times2\times2\times2\times2\times2\times3\times3}}{\sqrt{10000}}$
$\frac{2\times2\times2\times2\times3}{100}$
$=0.48$
$\sqrt{0.1764}=\sqrt{\frac{1764}{10000}}$
$=\frac{\sqrt{2\times2\times3\times3\times7\times7}}{\sqrt{10000}}$
$=\frac{2\times3\times7}{100}$
$=0.42$
$\frac{\sqrt{0.2304}\ +\sqrt{0.1764}}{\sqrt{0.2304}\ -\sqrt{0.1764}}=\frac{0.48+0.42}{0.48-0.42}$
$=\frac{0.9}{0.06}$
$=15$
View full question & answer→Question 903 Marks
Find the square root of the following by prime factorization.$24336$
AnswerResolving 24336 into prime factors, $24336 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 13 \times 13$
$\begin{array}{c|c}2& 24336 \\ \hline 2 & 12168 \\\hline 2&6084 \\\hline2&3042 \\\hline3&1521 \\\hline 3&507\\\hline13&169\\\hline13&13\\\hline&1\end{array}$
Grouping the factors into pairs of equal factors, $24336 = (2 \times 2) \times (2 \times 2) \times (3 \times 3) \times (13 \times 13)$
Taking one factor for each pair,
we get the square root of $24336 2 \times 2 \times 3 \times 13 = 156$
View full question & answer→Question 913 Marks
Find the square root of the following correct to three places of decimal:
$427$
AnswerWe 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 $20.664$ View full question & answer→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.
AnswerSince $71$ 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, $5929 = 7 \times 7 \times 11 \times 11$
Grouping them into pairs of equal factors, $5929 = (7 \times 7) \times (11 \times 11)$
The square root of $5929 =\sqrt{5929}=7\times11=77$
Hence, in the arrangement, there were $77$ rows of students.
View full question & answer→