Question 12 Marks
Find the length of each side of a cube if its volume is $512cm^3$.
AnswerGiven, volume of a cube $= 512cm^2$
$\mathrm{\Rightarrow (Side\ of\ a \ cube)^3= 8 × 8 × 8 = (8)^3}$
$\therefore$ Side of a cube $= 8\ cm$
Hence, the length of each side of the cube is $8\ cm$.
View full question & answer→Question 22 Marks
Find the square root of the following by long division method.$1.44$
AnswerWe have, $1.44$ $\ \ \ \ 1.2\\\begin{array}{c|c}\hline 1 & {\overline{1\\1}}\ {\overline{44}} \\ \hline22 & {44\\44}\\\hline&0\end{array}$ $\therefore\sqrt{1.44}=1.2$
View full question & answer→Question 32 Marks
Is $9720$ a perfect cube? If not, find the smallest number by which it should be divided to get a perfect cube.
AnswerIf we divide the number by $3 \times 3 \times 5$, then the prime factorisation of the quotient will not contain $3 \times 3 \times 5 = 45$.
$\begin{array}{c|c}2&9720\\\hline2&4860\\\hline2&2430\\\hline3&1215\\\hline3&405\\\hline3&135\\\hline3&45\\\hline3&15\\\hline5&5\\\hline&1 \end{array}$ $\therefore9720+45=\underline{2\times2\times2}\times\underline{3\times3\times3}$
$=216$ $=(6)^3$
Prime factors of $9720 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 3 \times 3 \times 5$
The prime factors $3$ and $5$ do not appear in group of triplets.
So, $9720$ is not a perfect cube.If we divide the number by $3 \times 3 \times 5,$
then the prime factorisation of the quotient will not contain $3 \times 3 \times 5 = 45.$
View full question & answer→Question 42 Marks
Find the square root of the following by long division method. $1369$
AnswerWe have, $1369$
$\ \ \ \ 37\\\begin{array}{c|c}\hline 3 & {\overline{13\\9}}\ {\overline{69}} \\ \hline67 & {469\\469}\\\hline&0\end{array}$
$\therefore\sqrt{1369}=37$
View full question & answer→Question 52 Marks
Find the greatest number of three digits that is a perfect square.
AnswerSince, the greatest number of three digits is $1000$.
Now, find the square root of $1000$ by long division method.
$\begin{array}{c|c} & 31 \\ \hline3 & \overline{10\\9}\ \overline{00}\\\hline61&{100\\61}\\\hline&39\\\end{array}$
Hence, the greatest number of three digits, i.e. a perfect square $= 1000 - 39 = 961$
View full question & answer→Question 62 Marks
Find the square root of the following by long division method.$5625$
AnswerWe have, $5625$
$\ \ \ \ 75\\\begin{array}{c|c}\hline 7 & {\overline{56\\49}}\ {\overline{25}} \\ \hline145 & {725\\725}\\\hline&0\end{array}$
$\therefore\sqrt{5625}=75$
View full question & answer→Question 72 Marks
A perfect square number has four digits, none of which is zero. The digits from left to right have values that are: even, even, odd, even. Find the number.
AnswerSuppose abed is a perfect square.
where, $a$ = even number $B$ = even number $C$ = odd number $D$ = even number
Hence, $8836$ is one of the number which satisfies the given condition.
View full question & answer→Question 82 Marks
Three numbers are in the ratio $1 : 2 : 3$ and the sum of their cubes is $4500$. Find the numbers.
AnswerLet the three numbers be $x, 2x$ and $3x$.
According to the question, $(\text{x})^3+(2\text{x})^2+(3\text{x})^3=45000$
$\Rightarrow\text{x}^3+8\text{x}^3+27\text{x}^3=45000$
$\Rightarrow36\text{x}^3=45000$
$\Rightarrow\text{x}^3=\frac{45000}{36}$
$\Rightarrow\text{x}^3=125$ [taking cube root on both sides]
$\Rightarrow\text{x}=\sqrt[3]{125}$
$\Rightarrow\text{x}=\sqrt[3]{5\times5\times5}$
$\Rightarrow\text{x}=5$
Hence, the numbers are $5, 2$ and $3 \times 5$, i.e. $5, 10$ and $15$.
View full question & answer→Question 92 Marks
What will be the number of unit squares on each side of a square graph paper if the total number of unit squares is $256$?
AnswerLet the number be $x$. Then, number of unit squares $= x × x = x^2$
But total number of unit squares $= 256$
$\therefore\text{x}^2=256$
$\Rightarrow\text{x}=\sqrt{256}=\sqrt{16\times16}=16$
Hence, the required number of squares is $16$.
View full question & answer→Question 102 Marks
How many square metres of carpet will be required for a square room of side $6.5m$ to be carpeted.
AnswerSide of a square room $= 6.5m$
To find area of square room
Area of carpet to be required = (Side of the square room)$^2= (6.5m)^2= 42.25m^2$
View full question & answer→Question 112 Marks
Using prime factorisation, find which of the following are perfect squares.$841$
AnswerPrime factors of $841 = (29 \times 29)$
$\begin{array}{c|c} 29 & 841 \\ \hline 29 & 29\\\hline&1 \end{array}$
As grouping, there is no unpaired factor left over. So, $841$ is a perfect square.
View full question & answer→Question 122 Marks
Using prime factorisation, find the cube roots of:
$512$
AnswerWe have, $512$
$\begin{array}{c|c}2 & 512 \\ \hline2 & 256\\\hline2&128\\\hline2&64\\\hline2&32\\\hline2&16\\\hline2&8\\\hline2&4\\\hline2&2\\\hline&1\end{array}$
Now, $512 =2\times2\times2\times2\times2\times2\times2\times2\times2$
$\therefore\sqrt[3]{512}=2\times2\times2$
$=8$
View full question & answer→Question 132 Marks
$13$ and $31$ is a strange pair of numbers such that their squares $169$ and $961$ are also mirror images of each other. Can you find two other such pairs?
Answer$i.$ Let a pair of numbers be $12$ and $21$.
Also, $(12)^2= 144$ and $(21)^2= 441$
$ii.$ Let a pair of numbers be $102$ and $201$.
Also, $(102)^2= 10404$ and $(201)^2= 40401$
View full question & answer→Question 142 Marks
Using prime factorisation, find the cube roots of:$2197$
Answerwe have, $2197$
$\begin{array}{c|c}13 & 2197 \\ \hline13 & 169\\\hline13&13\\\hline&1 \end{array}$
Now, $2197 =13\times13\times13$
$\therefore\sqrt[3]{2197}=13$
View full question & answer→Question 152 Marks
Using prime factorisation, find which of the following are perfect squares.$11250$
AnswerPrime factors of $11250 = 2 \times (3 \times 3) \times (5 \times 5) \times (5 \times 5)$
As grouping, $2$ has no pair.
$\begin{array}{c|c}2 & 11250 \\ \hline 3 & 5625\\\hline3&1875\\\hline5&625\\\hline5&125\\\hline5&25\\\hline5&5\\\hline5&1 \end{array}$
So, $11250$ is not a perfect square,
View full question & answer→Question 162 Marks
What is the least number that should be added to $6200$ to make it a perfect square?
Answer$41$ is the least number
$6200 + 41 = 6241$
$\sqrt{6241}=69$
Mark as brainliest answer.
View full question & answer→Question 172 Marks
Find the square root of the following by long division method.
$27.04$
AnswerWe have, $1.44$
$\ \ \ \ 5.2\\\begin{array}{c|c}\hline 5 & {\overline{27\\25}}\ {\overline{0.4}} \\ \hline102 & {204\\204}\\\hline&0\end{array}$
$\therefore\sqrt{27.04}=5.2$
View full question & answer→Question 182 Marks
Using prime factorisation, find which of the following are perfect squares.
$484$
AnswerPrime factors of $484 = (2 \times 2) \times (11 \times 11)$
As grouping, there is no unpaired factor left over.
$\begin{array}{c|c} 2 & 484 \\ \hline 2 & 242\\\hline11&121\\\hline&1 \end{array}$
So, $484$ is a perfect square.
View full question & answer→Question 192 Marks
Using prime factorisation, find the square roots of:$4761$
AnswerWe have,$4761$
$\begin{array}{c|c}3 & 4761 \\ \hline3 & 1587\\\hline23&529\\\hline23&23\\\hline&1\end{array}$
Now, $4761$$=\underline{3\times3}\times\underline{23\times23}$
$\therefore$ Square root of $4761$ $=\sqrt{4761}=3\times23=69$
View full question & answer→Question 202 Marks
Is $176$ a perfect square? If not, find the smallest number by which it should be multiplied to get a perfect square.
AnswerPrime factor of $176$ $\underline{2\times2}\times\underline{2\times2}\times11$
Here, $11$ has no pair.
So, it is not perfect square and $11$ is the smallest number by which $176$ should be multiplied to get a perfect square.
Then, $176 × 11 = 2^2× 2^2× 11 × 11$
$ = 2^2× 2^2× 11^2$
$= 1936 = (44)^2$
which is a perfect square of $44$.
hence, the smallest number is $11$ which it should be multiplied to get a perfect square.
View full question & answer→Question 212 Marks
Find the area of a square field if its perimeter is $96m$.
AnswerGiven, perimeter of square field $= 96m$
$\because$ Perimeter of square $= 4 \times $ $Side$
$\Rightarrow 4 \times $ $(Side)$ $= 96$
$\Rightarrow $ $Side$ $= 24m$
$\therefore$ Area of square field $= (Side)^2= (24)^2= 576m^2$
View full question & answer→Question 222 Marks
A General wishes to draw up his $7500$ soldiers in the form of a square. After arranging, he found out that some of them are left out. How many soldiers were left out?
AnswerGiven, total number of soldiers $= 7500$
$\begin{array}{c|c} & 86 \\ \hline8 & {\overline{75}\\64}\ {\overline{00}}\\\hline166&{1100\\996}\\\hline&104\end{array}$
Hence, $104$ soldiers were left out.
View full question & answer→Question 232 Marks
Using prime factorisation, find the square roots of:
$11025$
AnswerWe have, $11025$
$\begin{array}{c|c}3 & 11025 \\ \hline3 & 3675\\\hline5&1225\\\hline5&245\\\hline7&49\\\hline 7&7\\\hline&1\end{array}$
Now, $11025=\underline{3\times3}\times\underline{5\times5}\times\underline{7\times7}$
$\therefore$ Square root of $11025$ $=\sqrt{11025}=3\times5\times7=105$
View full question & answer→Question 242 Marks
Show that $500$ is not a perfect square.
AnswerResolving $500$ into prime factors, we have
$\begin{array}{c|c} 2 & 500 \\ \hline 2 & 250\\\hline5&125\\\hline5&25\\\hline5&5 \\\hline&1\end{array}$
$500 = 2 \times 2 \times 5 \times 5 \times 5$
Grouping the factors into pairs of equal factors, we are left with one factor $5$, which cannot be paired.
Hence, $500$ is not a perfect square.
View full question & answer→Question 252 Marks
Find the least number of four digits that is a perfect square.
AnswerSince, the least number of four digits is $1000$.
Now, find the square root of $1000$ by long division method.
$\begin{array}{c|c} & 32 \\ \hline3 & \overline{10}\ \overline{00\\9}\\\hline62&{100\\124}\\\hline&-24\\\end{array}$
Hence, the least number of four digits, i.e. a perfect square $= 1000 + 24 = 1024$
View full question & answer→Question 262 Marks
Put three different numbers in the circles so that when you add the numbers at the end of each line you always get a perfect square.
Answer$\because$ $6 + 19 = 25$ (perfect square) $19 + 30 = 49$ (perfect square) and $30 + 6 = 36$ (perfect square) 
View full question & answer→Question 272 Marks
Using prime factorisation, find which of the following are perfect squares.$729$
AnswerPrime factors of $729 = (3 \times 3) \times (3 \times 3) \times (3 \times 3)$
As grouping, there is no unpaired factor left over.
$\begin{array}{c|c}3 & 729 \\ \hline3 & 243\\\hline3&81\\\hline3&27\\\hline3&9\\\hline3&3\\\hline3&1 \end{array}$
So, $729$ is a perfect square.
View full question & answer→Question 282 Marks
Find the square root of $324$ by the method of repeated subtraction.
AnswerFrist, find the square root of $1385$ by long division method.
$\ \ \ \ 37\\\begin{array}{c|c}\hline 3 & {\overline{13\\9}}\ {\overline{85}} \\ \hline67 & {485\\469}\\\hline&16\end{array}$
Hence, the least number is $16$, which should be subtracted from $1385$ to get a perfect square and the required perfect square number $= 1385 - 16 1369$
$\therefore$ Square root of 1369 $=\sqrt{37\times37}=37$
View full question & answer→