Question 12 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. $7623$
AnswerResolving $7623$ into prime factors:
$7623$
$= 3 × 3 × 7 × 11 × 11$
$= 3^2 × 7 × 11^2$
Thus, to get a perfect square, the given number should be multiplied by $7.$
New number$ = (3^2× 7^2 × 11^2)$
$= (3 × 7 × 11)^2$
$= (231)^2$
Hence, the number whose square is the new number is $231.$
View full question & answer→Question 22 Marks
Using the prime factorisation method, find the following numbers are perfect squares: $441$
AnswerA perfect square can always be expressed as a product of equal factors.
Resolving into prime factors:
$441$
$= 19 × 9$
$= 7 × 7 × 3 × 3$
$= 7 × 3 × 7 × 3$
$= 21 × 21$
$= (21)^2$
Thus, $441$ is a perfect square.
View full question & answer→Question 32 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. $2925$
AnswerResolving $2925$ into prime factors:
$2925$
$= 3 \times 3 \times 5 \times 5 \times 13$
$= 3^2\times 5^2\times 13$
Thus, to get a perfect square, the given number should be multiplied by $13.$
New number $= (3^2\times 5^2\times 13^2)$
$= (3 \times 5 \times 13)^2$
$= (195)^2$
Hence, the number whose square is the new number is $195.$
View full question & answer→Question 42 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. $3380$
AnswerResolving $3380$ into prime factors:
$3380$
$= 2 \times 2 \times 5 \times 13 \times 13$
$= 2^2\times 5 \times 13^2$
Thus, to get a perfect square, the given number should be multiplied by $5.$
New number $= (2^2\times 5^2\times 13^2)$
$= (2 \times 5 \times 13)^2$
$= (130)^2$
Hence, the new number is the square of $130.$
View full question & answer→Question 52 Marks
Find the largest number of $2$ digits which is a perfect square.
AnswerThe first three digit number $(100)$ is a perfect square. Its square root is $10.$
The number before $10$ is $9.$
Square of $9 = (9)^2= 81$
Thus, the largest 2 digit number that is a perfect square is $81.$
View full question & answer→Question 62 Marks
Using the formula $(a-b)^2=\left(a^2-2 a b+b^2\right)$, evaluate:
$(891)^2$
Answer$(891)^2$
$=(900-9)^2 $
$ =(900)^2-2 \times 900 \times 9+(9)^2 $
$ =810000-16200+81 $
$ =793881$
View full question & answer→Question 72 Marks
Show that the following numbers is a perfect square. In case, find the number whose square is the given number: $2601$
AnswerA perfect square is a product of two perfectly equal numbers.
Resolving into prime factors:
$2601$
$= 9 × 289$
$= 3 × 3 × 17 × 17$
$= 3 × 17 × 3 × 17$
$= 51 × 51$
$= (51)^2$
Thus, $2601$ is the perfect square of $51.$
View full question & answer→Question 82 Marks
Write a pythagorean triplet whose smallest member is: $20$
AnswerFor every natural number $m>1,\left(2 m, m^2-1, m^2+1\right)$ is a Pythagorean triplet.
Putting $2m = 20,$
We get $m = 10$
Thus, we get the triplet $(20, 99, 10).$
View full question & answer→Question 92 Marks
Evaluate:
$\frac{\sqrt{1183}}{\sqrt{2023}}$
Answer $\frac{\sqrt{1183}}{\sqrt{2023}}$
$=\sqrt{\frac{1183}{2023}}$
$=\sqrt{\frac{1183\div7}{2023\div7}}$
$=\frac{\sqrt{169}}{\sqrt{289}}$
$=\frac{\sqrt{13\times13}}{\sqrt{17\times17}}$
$=\frac{13}{17}$
View full question & answer→Question 102 Marks
Write a pythagorean triplet whose smallest member is: $14$
AnswerFor every natural number $m>1,\left(2 m, m^2-1, m^2+1\right)$ is a Pythagorean triplet.
Putting $2m = 14,$
We get $m = 7$
Thus, we get the triplet $(14, 48, 50).$
View full question & answer→Question 112 Marks
Find the square root of number by using the method of prime factorisation: $225$
AnswerBy prime factorisation method: $225=3\times3\times5\times5$ $\therefore\sqrt{225}=(3\times5)=15$
View full question & answer→Question 122 Marks
Evaluate:$\sqrt{9.8596}$
Answer$\begin{array}{c|c} &3.14 \\ \hline 3 & 9.\ \overline{85}\ \overline{96}\\& -9\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \hline61 &\ \ \ \ 85\ \ \ \ \ \ \\ &-61\ \ \ \ \ \ \\ \hline624 &\ 2496 \\ &-2496\ \ \\ \hline &\ \times \end{array}$
$\therefore\sqrt{9.8596}=3.14$
View full question & answer→Question 132 Marks
Find the square root of number by using the method of prime factorisation: $11025$
AnswerBy prime factorisation method: $11025=3\times3\times5\times5\times7\times7$
$\therefore\sqrt{11025}=(3\times5\times7)=105$
View full question & answer→Question 142 Marks
Write a pythagorean triplet whose smallest member is:$ 6$
AnswerFor every natural number $m>1,\left(2 m, m^2-1, m^2+1\right)$ is a Pythagorean triplet.
Putting $2m = 6$
We get $m = 3$
Thus, we get the triplet $(6, 8, 10).$
View full question & answer→Question 152 Marks
Evaluate:
$94 × 106$
Answer$= (100 - 6) × (100 + 6)$
$= [(100)^2- (6)^2]$
$= (10000 - 36)$
$= 9964$
View full question & answer→Question 162 Marks
Show that the following numbers is a perfect square. In case, find the number whose square is the given number:
$7056$
AnswerA perfect square is a product of two perfectly equal numbers.
Resolving into prime factors:
$7056$
$= 12 × 588$
$= 12 × 7 × 84$
$= 12 × 7 × 12 × 7$
$= (12 × 7)^2$
$= (84)^2$
Thus, $7056$ is the perfect square of $84.$
View full question & answer→Question 172 Marks
Write a pythagorean triplet whose smallest member is: $16$
AnswerFor every natural number $m>1,\left(2 m, m^2-1, m^2+1\right)$ is a Pythagorean triplet.
Putting $2m = 16,$
We get $m = 8$
Thus, we get the triplet $(16, 63, 65).$
View full question & answer→Question 182 Marks
Evaluate:
$ (105)^2- (104)^2$
AnswerWe have,
$(n+1)^2-n^2=(n+1)+n$
Taking $n = 104$ and $(n + 1) = 105$
We get,
$(105)^2- (104)^2= (105 + 104) = 209$
View full question & answer→Question 192 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.
$4500$
AnswerResolving $4500$ into prime factors:
$4500$
$= 2 \times 2 \times 3 \times 3 \times 5 \times 5 \times 5$
$=2^2 \times 3^2 \times 5^2 \times 5$
Thus, to get a perfect square, the given number should be divided by $5.$
New number obtained $=\left(2^2 \times 3^2 \times 5^2\right)$
$= (2 \times 3 \times 5)^2$
$= (30)^2$
Hence, the new number is the square of $30.$
View full question & answer→Question 202 Marks
Evaluate:$\sqrt{1.69}$
Answer$\begin{array}{c|c} &1.3 \\ \hline 1 & \bar{1}\ \overline{.69}\\& -1\ \ \ \ \ \ \ \ \ \\ \hline23 &\ \ 69\\ &-69\\ \hline &\ \ \ \times \end{array}$
$\therefore\sqrt{1.69}=1.3$
View full question & answer→Question 212 Marks
Evaluate:$\sqrt{75.69}$
Answer$\begin{array}{c|c} &8.7 \\ \hline 8 & \overline{75}\ \overline{.69}\\&-64 \ \ \ \ \ \ \ \ \ \\ \hline167 &\ \ \ \ \ \ \ \ \ 1169\ \ \ \ \ \\ &\ \ \ \ \ -1169\ \ \ \ \ \ \\ \hline &\ \ \ \ \ \ \ \ \ \ \times \end{array}$
$\therefore\sqrt{75.69}=8.7$
View full question & answer→Question 222 Marks
Evaluate: $\sqrt{156.25}$
Answer $\begin{array}{c|c} &12.5 \\ \hline 1 & \overline{156}\ \overline{.25}\\& -1\ \ \ \ \ \ \ \ \ \ \ \ \ \\ \hline28 &\ 56\ \ \ \ \ \\ &-56\ \ \ \ \ \ \\ \hline5 &\ \ \ \ \ \ \ \ \ 25 \\ &\ \ \ \ \ \ -25 \\ \hline &\ \ \ \ \ \ \ \ \ \ \times \end{array}$
$\therefore\sqrt{156.25}=12.5$
View full question & answer→Question 232 Marks
Using the formula $(a-b)^2=\left(a^2-2 a b+b^2\right)$, evaluate:
$ (689)^2$
Answer$ (689)^2$
$ =(700-11)^2 $
$ =(700)^2-2 \times 700 \times 11+(11)^2 $
$ =490000-15400+121 $
$ =474721 $
View full question & answer→Question 242 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.
$3380$
AnswerResolving $3380$ into prime factors:
$3380$
$= 2 \times 2 \times 5 \times 13 \times 13$
$= 2^2\times 5 \times 13^2$
Thus, to get a perfect square, the given number should be divided by $5.$
New number obtained $= (2^2\times 13^2)$
$= (2 \times 13)^2$
$= (26)^2$
Hence, the new number is the square of $26.$
View full question & answer→Question 252 Marks
Using the prime factorisation method, find the following numbers are perfect squares:
$4225$
AnswerA perfect square can always be expressed as a product of equal factors.
Resolving into prime factors:
$4225$
$= 25 × 169$
$= 5 × 5 × 13 × 13$
$= 5 × 13 × 5 × 13$
$= 65 × 65$
$= (65)^2$
Thus, $4225$ is a perfect square.
View full question & answer→Question 262 Marks
Find the least number which must be subtracted from $2509$ to make it a perfect square.
AnswerFinding the square root of $2509$ by division we find that $9$ is left as remainder
$\begin{array}{c|c} & 50 \\ \hline 5 & \overline{25}\ \overline{09}\\& 25 \ \ \ \ \ \\ \hline100 &\ \ \ \ \ \ \ 09\\ &\ \ \ \ \ \ \ 00\\ \hline &\ \ \ \ \ \ \ 09 \end{array}$
$9$ must be subtracted to get the perfect square $100$
Least number to be subtracted $= 9$
View full question & answer→Question 272 Marks
Find the square root of number by using the method of prime factorisation: $17424$
AnswerBy prime factorisation method: $17424=2\times2\times2\times2\times3\times3\times11\times11$$\therefore\sqrt{17424}=(2\times2\times3\times11)=132$
View full question & answer→Question 282 Marks
Evaluate:
$78 × 82$
Answer$= (80 - 2) × (80 + 2)$
$= [(80)^2- (2)^2]$
$= (6400 - 4)$
$= 6396$
View full question & answer→Question 292 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.
$4056$
AnswerResolving $4056$ into prime factors:
$4056$
$= 2 × 2 × 2 × 3 × 13 × 13 $
$= 2^2x 2 × 3 × 13^2$
Thus, to get a perfect square, the given number should be divided by 6, which is a product of $2$ and $3.$
New number obtained $= (2^2× 13^2)$
$= (2 × 13)^2$
$= (26)^2$
Hence, the new number is the square of $26$.
View full question & answer→Question 302 Marks
Evaluate: $\sqrt{17956}$
Answer$\begin{array}{c|c} & 134 \\ \hline 1 & \bar1\ \overline{79}\ \overline{56}\\& 1\ \ \ \ \ \ \ \ \ \ \\ \hline23 &79\\ &69\ \\ \hline264 &1056\\ &1056\\\hline &\times \end{array}$
$\sqrt{17956}=134$
View full question & answer→Question 312 Marks
Evaluate: $\frac{\sqrt{80}}{\sqrt{405}}$
Answer$\frac{\sqrt{80}}{\sqrt{405}}$ $=\sqrt{\frac{80}{405}}$ $=\sqrt{\frac{16}{81}}$ $=\frac{\sqrt{16}}{\sqrt{81}}$ $=\frac{4}{9}$
View full question & answer→Question 322 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. $3332$
AnswerResolving $3332$ into prime factors:
$3332$
$= 2 \times 2 \times 7 \times 7 \times 17$
$= 2^2\times 7^2\times 17$
Thus, to get a perfect square, the given number should be multiplied by $17.$
New number $= (2^2\times 7^2\times 17^2)$
$= (2 \times 7 \times 17^2)$
$= (238)^2$
Hence, the new number is the square of $238.$
View full question & answer→Question 332 Marks
Evaluate:
$(38)^2- (37)^2$
AnswerWe have,
$(n + 1)^2- n^2= (n + 1) + n.$
Taking $n = 37$ and $(n + 1) = 38,$
We get,
$(38)^2- (37)^2= (38 + 37) = 75$
View full question & answer→Question 342 Marks
Find the value of using the column method:
$ (23)^2$
AnswerGiven number $23 = 20 + 3$ Here,
$a = 20$ and $b = 3$
|
$a^2$
|
$2ab$
|
$b^2$
|
|
$(20)^2= 400$
|
$2 × 20 × 3 = 120$
|
$(3)^2= 9$
|
$\therefore$ $(23)^2= (400 + 120 + 9) = 529$ View full question & answer→Question 352 Marks
Evaluate: $\sqrt{19600}$
Answer$\begin{array}{c|c} & 140 \\ \hline 1 & \bar1\ \overline{96}\ \overline{00}\\& 1\ \ \ \ \ \ \ \ \ \ \\ \hline24 &96\\ &96\ \\ \hline280 &00\\ &00\\\hline &\times \end{array}$
$\sqrt{19600}=140$
View full question & answer→Question 362 Marks
Using the formula $(a + b)^2= (a^2+ 2ab + b^2),$ evaluate:
$(310)^2$
Answer$ (310)^2=(300+10)^2 $
$ =(300)^2+2 \times 300 \times 10+(10)^2 $
$ =(90000+6000+100) $
$ =96100 $
View full question & answer→Question 372 Marks
Express $100$ as the sum of $10$ odd numbers.
AnswerWe know that $n^2$ is equal to the sum of first n odd numbers.
$100 = 10^2$
$=$ Sum of $10$ odd numbers $= (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19)$
View full question & answer→Question 382 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. $2156$
AnswerResolving $2156$ into prime factors:
$2156$
$= 2 \times 2 \times 7 \times 7 \times 11$
$= (2^2\times 7^2\times 11)$
Thus to get a perfect square, the given number should be multiplied by $11,$
New number $= (2^2\times 7^2\times 11^2)$
$= (2 \times 7 \times 11^2)$
$= (154)^2$
Hence, the new number is the square of $154.$
View full question & answer→Question 392 Marks
Evaluate: $\sqrt{7056}$
Answer$\begin{array}{c|c} & 84 \\ \hline 8 & \overline{70}\ \overline{56}\\& 64\ \ \ \ \ \\ \hline164 &656\\ &656\\ \hline &\times \end{array}$
$\sqrt{7056}=84$
View full question & answer→Question 402 Marks
Evaluate: $\sqrt{14161}$
Answer$\begin{array}{c|c} & 119 \\ \hline 1 & \bar1\ \overline{41}\ \overline{61}\\& 1\ \ \ \ \ \ \ \ \ \ \\ \hline21 &41\\ &21\ \\ \hline229 &2061\\ &2061\\\hline &\times \end{array}$
$\sqrt{14161}=119$
View full question & answer→Question 412 Marks
Evaluate:$\sqrt{10.0489}$
Answer$\begin{array}{c|c} &3.17 \\ \hline 3 & \overline{10}.\ \overline{04}\ \overline{89}\\& -9\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \hline61 &104\ \ \ \ \ \ \\ &-61\ \ \ \ \ \ \ \\ \hline627 &\ 4389 \\ &-4389\ \ \\ \hline &\ \times \end{array}$
$\therefore\sqrt{10.0489}=3.17$
View full question & answer→Question 422 Marks
Using the prime factorisation method, find the following numbers are perfect squares: $1176$
AnswerA perfect square can always be expressed as a product of equal factors.
Resolving into prime factors:
$1176$
$= 7 \times 168$
$= 7 \times 21 \times 8$
$= 7 \times 7 \times 3 \times 2 \times 2 \times 2$
$1176$ cannot be expressed as a product of two equal numbers.
Thus, $1176$ is not a perfect square.
View full question & answer→Question 432 Marks
Evaluate: $\sqrt{9025}$
Answer$\begin{array}{c|c} & 95 \\ \hline 9 & \overline{90}\ \overline{25}\\& 81\ \ \ \ \ \\ \hline185 &925\\ &925\\ \hline &\times \end{array}$
$\sqrt{9025}=95$
View full question & answer→Question 442 Marks
Using the formula $(a + b)^2= (a^2+ 2ab + b^2),$ evaluate:
$(508)^2$
Answer$ (508)^2=(500+8)^2 $
$ =(500)^2+2 \times 500 \times 8+(8)^2 $
$ =(250000+8000+64) $
$ =258064 $
View full question & answer→Question 452 Marks
Evaluate $\sqrt{3}$ up to two places of decimal.
Answer$\begin{array}{c|c} &1.732 \\ \hline 1 &3. \overline{00}\ \overline{00}\ \overline{00}\\& -1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \hline27 &200\ \ \ \ \ \ \ \ \ \ \ \\ &-189\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \hline343 &\ 1100\ \ \ \ \ \ \\ &-1029\ \ \ \ \ \ \ \ \\ \hline \hline3462 &\ \ \ \ \ \ \ 7100\ \ \ \ \ \ \\ &\ \ \ \ -6924\ \ \ \ \ \ \ \ \\ \hline&\ \ \ \ \ \ \ \ \ \ \ \ 176\ \ \ \ \ \ \ \ \end{array}$ $\therefore\sqrt{3}=1732=1.73$ (Correct up to two places of decimal)
View full question & answer→Question 462 Marks
Find the square root of number by using the method of prime factorisation:
$15876$
AnswerBy prime factorisation method: $15876=2\times2\times3\times3\times3\times3\times7\times7$
$\therefore\sqrt{15876}=(2\times3\times3\times7)=126$
View full question & answer→Question 472 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. $1575$
AnswerResolving $1575$ into prime factors:
$1575$
$= 3 \times 3 \times 5 \times 5 \times 7$
$= 3 \times 5 \times 7$
Thus, to get a perfect square, the given number should be divided by $7.$
New number obtained $= (3^2\times 5^2)$
$= (3 \times 5)^2$
$= (15)^2$
Hence, the new number is the square of $15.$
View full question & answer→Question 482 Marks
Show that the following numbers is a perfect square. In case, find the number whose square is the given number:
$ 1225$
AnswerA perfect square is a product of two perfectly equal numbers.
Resolving into prime factors:
$1225$
$= 25 × 49$
$= 5 × 5 × 7 × 7$
$= 5 × 7 × 5 × 7$
$= 35 × 35$
$= (35)^2$
Thus, $1225$ is the perfect square of $35.$
View full question & answer→Question 492 Marks
Evaluate: $\sqrt{\frac{121}{256}}$
Answer$\sqrt{\frac{121}{256}}$ $=\frac{\sqrt{121}}{\sqrt{256}}$ $=\sqrt{\frac{11\times11}{16\times16}}$ $=\frac{11}{16}$
View full question & answer→Question 502 Marks
Using the prime factorisation method, find the following numbers are perfect squares:
$576$
AnswerA perfect square can always be expressed as a product of equal factors.
Resolving into prime factors:
$576$
$= 64 × 9$
$= 8 × 8 × 3 × 3$
$= 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3$
$= 24 × 24$
$= (24)^2$
Thus, $576$ is a perfect square
View full question & answer→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$
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$2ab$
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$b^2$
|
|
$(30)^2= 900$
|
$2 \times 30 \times 5 = 300$
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$(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 $b = 6$
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$a^2$
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2ab
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$b^2$
|
|
$(90)^2= 8100$
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$2 \times 90 \times 6 = 300$
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$(6)^2= 36$
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$\therefore (96)^2 = (8100 + 1080 + 36) = 9216$ 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 $b = 2$
|
$a^2$
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2ab
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$b^2$
|
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$(50)^2= 2500$
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$2 × 50 × 2 = 200$
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$(2)^2= 4$
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$\therefore$ $(52)^2= (2500 + 200 + 4) = 2704$ 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→