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:
$9075$
$= 25 \times 363$
$= 5 \times 5 \times 3 \times 11 \times 11$
$= 55 \times 55 \times 3$
$9075$ 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
Answer By 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 $11.$
New number obtained $= (3^2\times 7^2)$
$= (3 \times 7)^2$
$= (21)^2$
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 $3.$
New number $= (3^2\times 5^2\times 11^2)$
$= (3 \times 5 \times 11)^2$
$= (165)^2$
Hence, the new number is square of $165.$
View full question & answer→Question 562 Marks
Evaluate:
$88 × 92$
Answer$= (90 - 2) × (90 + 2)$
$= [(90)^2- (2)^2]$
$= (8100 - 4)$
$= 8096$
View full question & answer→Question 572 Marks
Using the formula $(a + b)^2= (a^2+ 2ab + b^2),$ 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:
$(141)^2- (140)^2$
AnswerWe 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:
$11025$
AnswerA 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.$
New number $= (3^2\times 5^2\times 7^2)$
$= (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 × 25$
$= 9 × 25 × 25$
$= 3 × 3 × 5 × 5 × 5 × 5$
$= 3 × 5 × 5 × 3 × 5 × 5$
$= 75 × 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$
AnswerWe 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 $35 = 30\ +$ 5Here,
$a = 30$ and $b = 5$
|
$a^2$
|
$2ab$
|
$b^2$
|
|
$(30)^2= 900$
|
$2 \times 30 \times 5 = 300$
|
$(5)^2= 25$
|
$\therefore (35)^2= (900 + 300 + 25) = 1225$ 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 $= (2^2\times 2^2\times 3^2\times 3^2)$
$= (2 \times 2 \times 3 \times 3)^2$
$= (36)^2$
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 $= (3^2\times 5^2\times 11)$
$= (3 \times 5 \times 11)^2$
$= (165)^2$
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 $(a - b)^2= (a^2- 2ab + b^2),$ evaluate:
$(196)^2$
Answer$(196)^2$
$= (200 - 4)^2$
$= (200)^2 - 2 × 200 × 4 + (4)^2$
$= 40000 - 1600 + 16$
$= 38416$
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$=(5^2\times 11^2)$
$= (5 \times 11)^2$
$= (55)^2$
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
|
$b^2$
|
|
$(90)^2= 8100$
|
$2 \times 90 \times 6 = 300$
|
$(6)^2= 36$
|
$\therefore (96)^2 = (8100 + 1080 + 36) = 9216$ 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:
$(92)^2-(91)^2$
AnswerWe 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:
$8820$
$= 2 × 2 × 3 × 3 × 5 × 7 × 7$
$= 2^2× 3^2× 5 × 7^2$
Thus, to get a perfect square, the given number should be divided by $5.$
New number obtained$= (2^2× 3^2× 7^2)$
$= (2 × 3 × 7)^2$
$= (42)^2$
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 × 71$
Answer$= (70 - 1) × (70 + 1)$
$= [(70)^2- (1)^2]$
$= (4900 - 1)$
$= 4899$
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:
$(218)^2- (217)^2$
AnswerWe 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:
$8281$
$= 49 × 169$
$= 7 × 7 × 13 × 13$
$= 7 × 13 × 7 × 13$
$= (7 × 13)^2$
$= (91)^2$
Thus, $8281$ is the perfect square of $91.$
View full question & answer→