Question 512 Marks
Using the prime factorisation method, find the following numbers are perfect squares:
9075
AnswerA perfect square can always be expressed as a product of equal factors.
Resolving into prime factors:
9075
= 25 × 363
= 5 × 5 × 3 × 11 × 11
= 55 × 55 × 3
9075 is not a product of two equal numbers.
Thus, 9075 is not a perfect square.
View full question & answer→Question 522 Marks
Find the square root of number by using the method of prime factorisation:
441
AnswerBy prime factorisation method:$441=3\times3\times7\times7$
$\therefore\sqrt{441}=3\times7=21$
View full question & answer→Question 532 Marks
By what least number should the given number be divided to get a perfect square number? In case, find the number whose square is the new number.
$4851$
AnswerResolving 4851 into prime factors:
$4851$
$=3 \times 3 \times 7 \times 7 \times 11$
$=3^2 \times 7^2 \times 11$
Thus, to get a perfect square, the given number should be divided by 11.
$=(3 \times 7)^2$
$=(21)^2$
$\text { New number obtained }=\left(3^2 \times 7^2\right)$
Hence, the new number is the square of 21 .
View full question & answer→Question 542 Marks
Evaluate:
$\sqrt{576}$
Answer$\begin{array}{c|c} & 24 \\ \hline 2 & \bar{5}\ \overline{76}\\& 4\ \ \ \ \ \\ \hline44 &176\\ &176\\ \hline &\times \end{array}$
$\sqrt{576}=24$
View full question & answer→Question 552 Marks
By what least number should the given number be multiplied to get a perfect square number? In case, find the number whose square is the new number.
$9075$
AnswerResolving 9075 into prime factors:
$9075$
$=3 \times 5 \times 5 \times 11 \times 11$
$=3 \times 5^2 \times 11^2$
Thus, to get a perfect sovare the given number should be multiplied by 3 .
$\text { New number }=$
$=(3 \times 5 \times 11)^2$
$=(165)^2$
Hence, the new number is square of 165 .
View full question & answer→Question 562 Marks
Evaluate:
$88 \times 92$
Answer$=(90-2) \times(90+2)$
$= {\left[(90)^2-(2)^2\right] }$
$= (8100-4)$
$= 8096$
View full question & answer→Question 572 Marks
Using the formula $(a+b)^2=\left(a^2+2 a b+b^2\right)$, evaluate:
$(630)^2$
Answer$(630)^2$
$=(600+30)^2$
$=(600)^2+2 \times 600 \times 30+(30)^2$
$=(360000+36000+900)$
$=396900$
View full question & answer→Question 582 Marks
Evaluate:
$(141)^2-(140)^2$
Answer
We have,
$(n+1)^2-n^2=(n+1)+n$
Taking $n=140$ and $(n+1)=141$
We get,
$(141)^2-(140)^2=(141+140)=281$
View full question & answer→Question 592 Marks
Using the prime factorisation method, find the following numbers are perfect squares:
$11025$
Answer
A perfect square can always be expressed as a product of equal factors.
Resolving into prime factors:
$11025$
$=441 \times 25$
$=49 \times 9 \times 5 \times 5$
$=7 \times 7 \times 3 \times 3 \times 5 \times 5$
$=7 \times 5 \times 3 \times 7 \times 5 \times 3$
$=105 \times 105$
$=(105)^2$
Thus, 11025 is a perfect square.
View full question & answer→Question 602 Marks
Find the square root of number by using the method of prime factorisation:
9216
AnswerBy prime factorisation method: $9216=2\times2\times2\times2\times2\times2\times2\times2\times2\times2\times3\times3$$\therefore\sqrt{ 9216}=(2\times2\times2\times2\times2\times3)=96$
View full question & answer→Question 612 Marks
By what least number should the given number be multiplied to get a perfect square number? In case, find the number whose square is the new number.
$3675$
AnswerResolving 3675 into prime factors:
$3675$
$=3 \times 5 \times 5 \times 7 \times 7$
Thus, to get a perfect square the given number should be multiplied by 3 .
$\text { New number }=\left(3^2 \times 5^2 \times 7^2\right)$
$=(3 \times 5 \times 7)^2$
$=(105)^2$
Hence, the new number is the square of 105.
View full question & answer→Question 622 Marks
Using the prime factorisation method, find the following numbers are perfect squares:
$5625$
AnswerA perfect square can always be expressed as a product of equal factors.
Resolving into prime factors:
$5625$
$=225 \times 25$
$=9 \times 25 \times 25$
$=3 \times 3 \times 5 \times 5 \times 5 \times 5$
$=3 \times 5 \times 5 \times 3 \times 5 \times 5$
$=75 \times 75$
$=(75)^2$
Thus, 5625 is a perfect square.
View full question & answer→Question 632 Marks
Evaluate:
$\sqrt{11449}$
Answer$\begin{array}{c|c} & 107 \\ \hline 1 & \bar1\ \overline{14}\ \overline{49}\\& 1\ \ \ \ \ \ \ \ \ \ \\ \hline207 &1449\\ &1449\\ \hline &\times \end{array}$
$\sqrt{11449}=107$
View full question & answer→Question 642 Marks
Find the square root of number by using the method of prime factorisation:
1249
AnswerBy prime factorisation method:$1249=2\times2\times2\times2\times 3 \times 3\times 3\times 3$
$\therefore\sqrt{1296}=2\times2\times3\times3=36$
View full question & answer→Question 652 Marks
Evaluate:
$\sqrt{1.0816}$
Answer$\begin{array}{c|c} &1.04 \\ \hline 1 & 1.\ \overline{08}\ \overline{16}\\&-1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \hline204 &\ 0816\\ &\ 0816\\ \hline &\ \ \ \ \times \end{array}$
$\therefore\sqrt{1.0816}=1.04$
View full question & answer→Question 662 Marks
Evaluate:
$(75)^2-(74)^2$
Answer
We have,
$(n+1)^2-n^2=(n+1)+n$
Taking $n =74$ and $( n +1)=75$
We get,
$(75)^2-(74)^2=(75+74)=149$
View full question & answer→Question 672 Marks
Find the square root of number by using the method of prime factorisation:
8100
AnswerBy prime factorisation method: $8100=2\times2\times3\times3\times3\times3\times5\times5$$\therefore\sqrt{ 8100}=(2\times3\times3\times5)=90$
View full question & answer→Question 682 Marks
Find the value of using the column method:
$(35)^2$
AnswerGiven number $35=30+5$
Here,
$a =30$ and $b =5$
| $a ^2$ |
$2ab$
|
$b^2$ |
| $(30)^2=900$ |
$2 × 30 × 5 = 300$
|
$(5)^2=25$ |
$\therefore(35)^2=(900+300+25)=1225$ View full question & answer→Question 692 Marks
By what least number should the given number be divided to get a perfect square number? In case, find the number whose square is the new number.
$7776$
AnswerResolving 7776 into prime factors:
$7776$
$=2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3$
$=2^2 \times 2^2 \times 2 \times 3^2 \times 3^2 \times 3$
Thus, to get a perfect square, the given number should be divided by 2 and 3 .
New number obtained $=\left(2^2 \times 2^2 \times 3^2 \times 3^2\right)$
$=(2 \times 2 \times 3 \times 3)^2$
$=(36)^2$
Hence, the new number is the square of 36 .
View full question & answer→Question 702 Marks
Find the square root of number by using the method of prime factorisation:
2025
AnswerBy prime factorisation method:$2025=3\times3\times3\times3\times5\times5$
$\therefore\sqrt{2025}=3\times3\times5=45$
View full question & answer→Question 712 Marks
Show that the following numbers is a perfect square. In case, find the number whose square is the given number:
$5929$
AnswerA perfect square is a product of two perfectly equal numbers.
Resolving into prime factors:
$5929$
$=11 \times 539$
$=11 \times 7 \times 77$
$=11 \times 7 \times 11 \times 7$
$=77 \times 77$
$=(77)^2$
Thus, $5929$ is the perfect square of $77.$
View full question & answer→Question 722 Marks
Evaluate:
$\sqrt{4489}$
Answer$\begin{array}{c|c} & 67 \\ \hline 6 & \overline{44}\ \overline{89}\\& 36\ \ \ \ \ \\ \hline127 &889\\ &889\\ \hline &\times \end{array}$
$\sqrt{4489}=67$
View full question & answer→Question 732 Marks
Find the square root of number by using the method of prime factorisation:
7056
AnswerBy prime factorisation method: $7056=2\times2\times2\times2\times3\times3\times7\times7$$\therefore\sqrt{7056}=(2\times2\times3\times7)=84$
View full question & answer→Question 742 Marks
Evaluate:
$\sqrt{0.2916}$
Answer$\begin{array}{c|c} &0.54 \\ \hline 5 & 0.\ \overline{29}\ \overline{16}\\& -25\ \ \ \ \\ \hline104 &\ \ \ \ \ \ 416\ \\ &-\ 416\\ \hline &\ \ \ \ \ \times \end{array}$
$\therefore\sqrt{0.2916}=0.54$
View full question & answer→Question 752 Marks
Find the largest number of $3$ digits which is a perfect square.
AnswerThe largest 3 digit number is 999 .
The number whose square is 999 is 31.61 .
Thus, the square of any number greater than 31.61 will be a 4 digit number.
Therefore, the square of 31 will be the greatest 3 digit perfect square.
$31^2=31 \times 31=961$
View full question & answer→Question 762 Marks
Evaluate:
$\sqrt{10404}$
Answer$\begin{array}{c|c} & 102 \\ \hline 1 & \bar1\ \overline{04}\ \overline{04}\\& 1\ \ \ \ \ \ \ \ \ \ \\ \hline202 &0404\\ &404\\ \hline &\times \end{array}$
$\sqrt{10404}=102$
View full question & answer→Question 772 Marks
Evaluate:
$\sqrt{1444}$
Answer$\begin{array}{c|c} & 38 \\ \hline 3 & \overline{14}\ \overline{44}\\& 9\ \ \ \ \ \\ \hline68 &544\\ &544\\ \hline &\times \end{array}$
$\sqrt{1444}=38$
View full question & answer→Question 782 Marks
By what least number should the given number be multiplied to get a perfect square number? In case, find the number whose square is the new number.
$2475$
AnswerResolving 2475 into prime factors:
$2475$
$=3 \times 3 \times 5 \times 5 \times 11$
$=3^2 \times 5^2 \times 11$
Thus, to get a perfect square, the given number should be multiplied by 11 .
New number $=\left(3^2 \times 5^2 \times 11\right)$
$=(3 \times 5 \times 11)^2$
$=(165)^2$
Hence, the new number is the square of 165.
View full question & answer→Question 792 Marks
Find the square root of number by using the method of prime factorisation:
729
AnswerBy prime factorisation method:$729=3\times3\times3 \times 3 \times 3 \times 3$
$\therefore\sqrt{729}=3\times3\times3=27$
View full question & answer→Question 802 Marks
Evaluate:
$\sqrt{\frac{64}{225}}$
Answer$\sqrt{\frac{64}{225}}$
$=\frac{\sqrt{64}}{\sqrt{225}}$
$=\sqrt{\frac{8\times8}{15\times15}}$
$=\frac{8}{15}$
View full question & answer→Question 812 Marks
Find the square root of number by using the method of prime factorisation:
4096
AnswerBy prime factorisation method: $4096=2\times2\times2\times2\times2\times2\times2\times2\times2\times2\times2\times2$$\therefore\sqrt{4096}=(2\times2\times2\times2\times2\times2)=64$
View full question & answer→Question 822 Marks
Evaluate:
$\sqrt{\frac{16}{81}}$
Answer$\sqrt{\frac{16}{81}}$
$=\frac{\sqrt{16}}{\sqrt{81}}$
$=\sqrt{\frac{4\times4}{9\times9}}$
$=\frac{4}{9}$
View full question & answer→Question 832 Marks
Using the formula $(a-b)^2=\left(a^2-2 a b+b^2\right)$, evaluate:
$(196)^2$
Answer$(196)^2$
$=(200-4)^2$
$=(200)^2-2 \times 200 \times 4+(4)^2$
$=40000-1600+16$
$=38416$
View full question & answer→Question 842 Marks
Evaluate:
$\sqrt{92416}$
Answer$\begin{array}{c|c} & 304 \\ \hline 3 & \bar9\ \overline{24}\ \overline{16}\\& 9\ \ \ \ \ \ \ \ \ \ \\ \hline604 &2416\\ &2416\\ \hline &\times \end{array}$
$\sqrt{92416}=304$
View full question & answer→Question 852 Marks
By what least number should the given number be divided to get a perfect square number? In case, find the number whose square is the new number.
$9075$
AnswerResolving 9075 into prime factors:
$9075$
$=3 \times 5 \times 5 \times 11 \times 11$
$=3 \times 5^2 \times 11^2$
Thus, to get a perfect square, the given number should be divided by 3 .
New number obtained $=\left(5^2 \times 11^2\right)$
$=(5 \times 11)^2$
$=(55)^2$
Hence, the new number is the square of 55.
View full question & answer→Question 862 Marks
Find the value of using the column method:
$(96)^2$
AnswerGiven number $96=90+6$
Here,
$a =90$ and $b =6$
| $a ^2$ |
$2ab$
|
$b^2$
|
| $(90)^2=8100$ & $2 2 a b$ |
2 × 90 × 6 = 300
|
(6)2= 36
|
$\therefore(96)^2=(8100+1080+36)=9216$ View full question & answer→Question 872 Marks
Using the prime factorisation method, find the following numbers are perfect squares:
1089
AnswerA perfect square can always be expressed as a product of equal factors.
Resolving into prime factors:
$1089$
$=9 \times 121$
$=3 \times 3 \times 11 \times 11$
$=3 \times 11 \times 3 \times 11$
$=33 \times 33$
$=(33)^2$
Thus, 1089 is a perfect square.
View full question & answer→Question 882 Marks
Evaluate:
$(92)^2-(91)^2$
Answer
We have,
$(n+1)^2-n^2=(n+1)+n$
Taking $n =91$ and $( n +1)=92$
We get,
$(92)^2-(91)^2=(92+91)=183$
View full question & answer→Question 892 Marks
By what least number should the given number be divided to get a perfect square number? In case, find the number whose square is the new number.
$8820$
AnswerResolving 8820 into prime factors:
$8820$
$=2 \times 2 \times 3 \times 3 \times 5 \times 7 \times 7$
$=2^2 \times 3^2 \times 5 \times 7^2$
Thus, to get a perfect square, the given number should be divided by 5 .
New number obtained $=\left(2^2 \times 3^2 \times 7^2\right)$
$=(2 \times 3 \times 7)^2$
$=(42)^2$
Hence, the new number is the square of 42 .
View full question & answer→Question 902 Marks
Evaluate:$\sqrt{33.64}$
Answer$\begin{array}{c|c} &5.8 \\ \hline 5 & \overline{33}\ \overline{.64}\\& -25\ \ \ \ \ \ \ \ \ \\ \hline108 &\ 864\\ &-864\ \ \\ \hline &\ \ \ \times \end{array}$
$\therefore\sqrt{33.64}=5.8$
View full question & answer→Question 912 Marks
Evaluate:
$69 \times 71$
Answer$=(70-1) \times(70+1)$
$= & {\left[(70)^2-(1)^2\right] }$
$= & (4900-1)$
$= & 4899$
View full question & answer→Question 922 Marks
Express 81 as the sum of 9 odd numbers.
AnswerWe know that $n ^2$ is equal to the sum of first n odd numbers.
$81=9^2$
$=$ Sum of 9 odd numbers $=(1+3+5+7+9+11+13+15+17)$
View full question & answer→Question 932 Marks
Evaluate:
$\sqrt{6241}$
Answer$\begin{array}{c|c} & 79 \\ \hline 7 & \overline{62}\ \overline{41}\\& 49\ \ \ \ \ \\ \hline149 &1341\\ &1341\\ \hline &\times \end{array}$
$\sqrt{6241}=79$
View full question & answer→Question 942 Marks
Find the value of using the column method:
$(52)^2$
AnswerGiven number $52=50+2$
Here,
$a =50$ and $b =2$$$
| $a ^2$ |
$2ab$
|
$b^2$
|
| $(50)^2=2500$ |
$2 × 50 × 2 = 200$
|
$(2)^2=4$
|
$\therefore(52)^2=(2500+200+4)=2704$ View full question & answer→Question 952 Marks
Evaluate:
$(218)^2-(217)^2$
Answer
We have,
$(n+1)^2-n^2=(n+1)+n$
Taking $n =140$ and $( n +1)=141$
We get,
$(218)^2-(217)^2=(218+217)=435$
View full question & answer→Question 962 Marks
Show that the following numbers is a perfect square. In case, find the number whose square is the given number:
$8281$
AnswerA perfect square is a product of two perfectly equal numbers.
Resolving into prime factors:
$8281$
$=49 \times 169$
$=7 \times 7 \times 13 \times 13$
$=7 \times 13 \times 7 \times 13$
$=(7 \times 13)^2$
$=(91)^2$
Thus, 8281 is the perfect square of 91.
View full question & answer→