Q10. Find the square root of 176.252176
Sol. By using long division method, we have\begin{array}{l|l} & 13.276 \\ \hline 1 & 1 ~ \overline{76.} ~ \overline{25} ~ \overline{21} ~ \overline{76} \\ & 1 \\ \hline 13 & ~76 \\ & ~69 \\ \hline 262 & ~~~725 \\ & ~~~524 \\ \hline 2647 & ~~~~~20121 \\ & ~~~~~18529 \\ \hline 26546 & ~~~~~159276 \\ & ~~~~~159276 \\ \hline & ~~~~~ 0 \\ \end{array}$$ \therefore \text {The Square root of } 176.252176 = 13.276 $$ $$ \sqrt {176.252176} = 13.276 $$Q11. Find the square root of 9998.0001
Sol. By using long division method, we have\begin{array}{l|l} & 99.99 \\ \hline 9 & \overline{99} ~ \overline{98.} ~ \overline{00} ~ \overline{01} \\ & 81 \\ \hline 189 & ~1898 \\ & ~1701 \\ \hline 1989 & ~~~19700 \\ & ~~~17901 \\ \hline 19989 & ~~~~~179901 \\ & ~~~~~179901 \\ \hline & ~~~~~ 0 \\ \end{array}Therefore, the Square root of 9998.0001 = 99.99
$$ \sqrt {9998.0001} = 99.99 $$Q12. Find the square root of 0.00038809
Sol. By using long division method, we have\begin{array}{l|l} & 0.0197 \\ \hline 0 & 0. ~ \overline{00} ~ \overline{03} ~ \overline{88} ~ \overline{09} \\ & 0 \\ \hline 1 & ~03 \\ & ~1 \\ \hline 29 & ~~~288 \\ & ~~~261 \\ \hline 387 & ~~~~~2709 \\ & ~~~~~2709 \\ \hline & ~~~~~ 0 \\ \end{array}Therefore, the Square root of 0.00038809 = 0.0197
$$ \sqrt {0.00038809} = 0.0197 $$Q13. What is that fraction which when multiplied by itself gives 227.798649 ?
Sol. Lets assume x be the fraction which when multiplied by itself give 227.798649
$$ \Rightarrow \quad x^{2} = 227.798649 $$ $$ \Rightarrow \quad x = \sqrt{227.798649} $$ By using long division method, we have
\begin{array}{l|l} & 15.093 \\ \hline 1 & 2 ~ \overline{27.} ~ \overline{79} ~ \overline{86} ~ \overline{49} \\ & 1 \\ \hline 25 & ~127 \\ & ~125 \\ \hline 300 & ~~~279 \\ & ~~~0 \\ \hline 3009 & ~~~~~27986 \\ & ~~~~~27081 \\ \hline 30183 & ~~~~~~~~90549 \\ & ~~~~~~~~90549 \\ \hline & ~~~~~~~~ 0 \\ \end{array}Therefore, 15.093 is the required fraction.
Q14. The area of a square playground is 256.6404 square metres. Find the length of one side of the playground.
$$ \text {Sol. Given that area of square playground }= 256.6404 m^{2} $$ Lets assume x meters be the side length of the playground.
$$ \Rightarrow \quad x^{2} = 256.6404 $$ $$ \Rightarrow \quad x = \sqrt{256.6404} $$ By using long division method, we have
\begin{array}{l|l} & 16.02 \\ \hline 1 & 2 ~ \overline{56.} ~ \overline{64} ~ \overline{04} \\ & 1 \\ \hline 26 & ~156 \\ & ~156 \\ \hline 320 & ~~~64 \\ & ~~~0 \\ \hline 3202 & ~~~~~6404 \\ & ~~~~~6404 \\ \hline & ~~~~~~~~ 0 \\ \end{array}Therefore, length of one side of playground is 16.02 m.
Q15. What is the fraction which when multiplied by itself gives 0.00053361 ?
Sol. Lets assume x be the fraction which when multiplied by itself give 0.00053361
$$ \Rightarrow \quad x^{2} = 0.00053361 $$ $$ \Rightarrow \quad x = \sqrt{0.00053361} $$ By using long division method, we have
\begin{array}{l|l} & 0.0231 \\ \hline 0 & 0. ~ \overline{00} ~ \overline{05} ~ \overline{33} ~ \overline{61} \\ & 0 \\ \hline 2 & ~05 \\ & ~4 \\ \hline 43 & ~~~133 \\ & ~~~129 \\ \hline 461 & ~~~~~461 \\ & ~~~~~461 \\ \hline & ~~~~~ 0 \\ \end{array}Therefore, 0.0231 is the required fraction.
Q16. Simplify:
$$ (i) \quad \frac{\sqrt{59.29}-\sqrt{5.29}}{\sqrt{59.29}+\sqrt{5.29}} $$ $$ (ii) \quad \frac{\sqrt{0.2304}+\sqrt{0.1764}}{\sqrt{0.2304}-\sqrt{0.1764}} $$Sol. (i) By using long division method, we have
\begin{array}{l|l} & 77 \\ \hline 7 & \overline{59} ~ \overline{29} \\ & 49 \\ \hline 147 & ~1029 \\ & ~1029 \\ \hline & ~~~~~ 0 \\ \end{array}$$ \therefore \quad \sqrt{5929} = 77 $$ $$ \text {And, } \sqrt{59.29} = 7.7 $$\begin{array}{l|l} & 23 \\ \hline 2 & \overline{5} ~ \overline{29} \\ & 4 \\ \hline 43 & ~129 \\ & ~129 \\ \hline & ~~~~~ 0 \\ \end{array}$$ \therefore \quad \sqrt{529} = 23 $$ $$ \text {And, } \sqrt{5.29} = 2.3 $$ $$ \frac{\sqrt{59.29}-\sqrt{5.29}}{\sqrt{59.29}+\sqrt{5.29}} $$ $$ =\frac{7.7-2.3}{7.7+2.3} $$ $$ =\frac{5.4}{10} $$ $$ =0.54 $$(ii) By using long division method, we have
\begin{array}{l|l} & 48 \\ \hline 4 & \overline{23} ~ \overline{04} \\ & 16 \\ \hline 88 & ~704 \\ & ~704 \\ \hline & ~~~~~ 0 \\ \end{array}$$ \therefore \quad \sqrt{2304} = 48 $$ $$ \text {And, } \sqrt{.2304} = 0.48 $$\begin{array}{l|l} & 42 \\ \hline 4 & \overline{17} ~ \overline{64} \\ & 16 \\ \hline 82 & ~164 \\ & ~164 \\ \hline & ~~~~~ 0 \\ \end{array}$$ \therefore \quad \sqrt{529} = 42 $$ $$ \text {And, } \sqrt{5.29} = 0.42 $$ $$ \frac{\sqrt{0.2304}+\sqrt{0.1764}}{\sqrt{0.2304}-\sqrt{0.1764}} $$ $$ =\frac{0.48+0.42}{0.48-0.42} $$ $$ =\frac{0.9}{0.06} $$ $$ =15 $$$$ \text {Q17. Evaluate } \sqrt{50625} \text { and hence find the value of } \sqrt{506.25}+\sqrt{5.0625} $$Sol. (i) By using long division method, we have
\begin{array}{l|l} & 225 \\ \hline 2 & 5 ~ \overline{06} ~ \overline{25} \\ & 4 \\ \hline 42 & ~106 \\ & ~84 \\ \hline 445 & ~~~2225 \\ & ~~~2225 \\ \hline & ~~~~~ 0 \\ \end{array}$$ \therefore \quad \sqrt{50625} = 225 $$ $$ \Rightarrow \quad \sqrt{506.25} = \sqrt{\frac{50625}{100}} = \frac {225}{10} = 22.5 $$$$ \Rightarrow \quad \sqrt{5.0625} = \sqrt{\frac{50625}{10000}} = \frac {225}{100} = 2.25 $$Hence,
$$ \sqrt{506.25}+\sqrt{5.0625} = 22.5 + 2.25 = 24.75 $$$$ \text {Q18. Find the value of } \sqrt{103.0225} \text { and hence find the value of } $$ $$ (i) \quad \sqrt{10302.25} $$ $$ (ii) \quad \sqrt{1.030225} $$Sol.
\begin{array}{l|l} & 1015 \\ \hline 1 & 1 ~ \overline{03} ~ \overline{02} ~ \overline{25} \\ & 1 \\ \hline 20 & ~03 \\ & ~0 \\ \hline 201 & ~~~302 \\ & ~~~201 \\ \hline 2025 & ~~~~~10125 \\ & ~~~~~10125 \\ \hline & ~~~~~~ 0 \\ \end{array}$$ \therefore \quad \sqrt{1030225} = 1015 $$ $$ \Rightarrow \quad \sqrt{103.0225} =\sqrt{\frac{1030225}{10000}}= \frac{1015}{100} = 10.15 $$$$ (i) \quad \sqrt{10302.25} $$ $$ \Rightarrow \quad \sqrt{10302.25} = \sqrt{\frac{1030225}{100}} = \frac {1015}{10} = 101.5 $$$$ (ii) \quad \sqrt{1.030225} $$ $$ \Rightarrow \quad \sqrt{1.030225} = \sqrt{\frac{1030225}{1000000}} = \frac {1015}{1000} = 1.015 $$
Sol. By using long division method, we have\begin{array}{l|l} & 13.276 \\ \hline 1 & 1 ~ \overline{76.} ~ \overline{25} ~ \overline{21} ~ \overline{76} \\ & 1 \\ \hline 13 & ~76 \\ & ~69 \\ \hline 262 & ~~~725 \\ & ~~~524 \\ \hline 2647 & ~~~~~20121 \\ & ~~~~~18529 \\ \hline 26546 & ~~~~~159276 \\ & ~~~~~159276 \\ \hline & ~~~~~ 0 \\ \end{array}$$ \therefore \text {The Square root of } 176.252176 = 13.276 $$ $$ \sqrt {176.252176} = 13.276 $$Q11. Find the square root of 9998.0001
Sol. By using long division method, we have\begin{array}{l|l} & 99.99 \\ \hline 9 & \overline{99} ~ \overline{98.} ~ \overline{00} ~ \overline{01} \\ & 81 \\ \hline 189 & ~1898 \\ & ~1701 \\ \hline 1989 & ~~~19700 \\ & ~~~17901 \\ \hline 19989 & ~~~~~179901 \\ & ~~~~~179901 \\ \hline & ~~~~~ 0 \\ \end{array}Therefore, the Square root of 9998.0001 = 99.99
$$ \sqrt {9998.0001} = 99.99 $$Q12. Find the square root of 0.00038809
Sol. By using long division method, we have\begin{array}{l|l} & 0.0197 \\ \hline 0 & 0. ~ \overline{00} ~ \overline{03} ~ \overline{88} ~ \overline{09} \\ & 0 \\ \hline 1 & ~03 \\ & ~1 \\ \hline 29 & ~~~288 \\ & ~~~261 \\ \hline 387 & ~~~~~2709 \\ & ~~~~~2709 \\ \hline & ~~~~~ 0 \\ \end{array}Therefore, the Square root of 0.00038809 = 0.0197
$$ \sqrt {0.00038809} = 0.0197 $$Q13. What is that fraction which when multiplied by itself gives 227.798649 ?
Sol. Lets assume x be the fraction which when multiplied by itself give 227.798649
$$ \Rightarrow \quad x^{2} = 227.798649 $$ $$ \Rightarrow \quad x = \sqrt{227.798649} $$ By using long division method, we have
\begin{array}{l|l} & 15.093 \\ \hline 1 & 2 ~ \overline{27.} ~ \overline{79} ~ \overline{86} ~ \overline{49} \\ & 1 \\ \hline 25 & ~127 \\ & ~125 \\ \hline 300 & ~~~279 \\ & ~~~0 \\ \hline 3009 & ~~~~~27986 \\ & ~~~~~27081 \\ \hline 30183 & ~~~~~~~~90549 \\ & ~~~~~~~~90549 \\ \hline & ~~~~~~~~ 0 \\ \end{array}Therefore, 15.093 is the required fraction.
Q14. The area of a square playground is 256.6404 square metres. Find the length of one side of the playground.
$$ \text {Sol. Given that area of square playground }= 256.6404 m^{2} $$ Lets assume x meters be the side length of the playground.
$$ \Rightarrow \quad x^{2} = 256.6404 $$ $$ \Rightarrow \quad x = \sqrt{256.6404} $$ By using long division method, we have
\begin{array}{l|l} & 16.02 \\ \hline 1 & 2 ~ \overline{56.} ~ \overline{64} ~ \overline{04} \\ & 1 \\ \hline 26 & ~156 \\ & ~156 \\ \hline 320 & ~~~64 \\ & ~~~0 \\ \hline 3202 & ~~~~~6404 \\ & ~~~~~6404 \\ \hline & ~~~~~~~~ 0 \\ \end{array}Therefore, length of one side of playground is 16.02 m.
Q15. What is the fraction which when multiplied by itself gives 0.00053361 ?
Sol. Lets assume x be the fraction which when multiplied by itself give 0.00053361
$$ \Rightarrow \quad x^{2} = 0.00053361 $$ $$ \Rightarrow \quad x = \sqrt{0.00053361} $$ By using long division method, we have
\begin{array}{l|l} & 0.0231 \\ \hline 0 & 0. ~ \overline{00} ~ \overline{05} ~ \overline{33} ~ \overline{61} \\ & 0 \\ \hline 2 & ~05 \\ & ~4 \\ \hline 43 & ~~~133 \\ & ~~~129 \\ \hline 461 & ~~~~~461 \\ & ~~~~~461 \\ \hline & ~~~~~ 0 \\ \end{array}Therefore, 0.0231 is the required fraction.
Q16. Simplify:
$$ (i) \quad \frac{\sqrt{59.29}-\sqrt{5.29}}{\sqrt{59.29}+\sqrt{5.29}} $$ $$ (ii) \quad \frac{\sqrt{0.2304}+\sqrt{0.1764}}{\sqrt{0.2304}-\sqrt{0.1764}} $$Sol. (i) By using long division method, we have
\begin{array}{l|l} & 77 \\ \hline 7 & \overline{59} ~ \overline{29} \\ & 49 \\ \hline 147 & ~1029 \\ & ~1029 \\ \hline & ~~~~~ 0 \\ \end{array}$$ \therefore \quad \sqrt{5929} = 77 $$ $$ \text {And, } \sqrt{59.29} = 7.7 $$\begin{array}{l|l} & 23 \\ \hline 2 & \overline{5} ~ \overline{29} \\ & 4 \\ \hline 43 & ~129 \\ & ~129 \\ \hline & ~~~~~ 0 \\ \end{array}$$ \therefore \quad \sqrt{529} = 23 $$ $$ \text {And, } \sqrt{5.29} = 2.3 $$ $$ \frac{\sqrt{59.29}-\sqrt{5.29}}{\sqrt{59.29}+\sqrt{5.29}} $$ $$ =\frac{7.7-2.3}{7.7+2.3} $$ $$ =\frac{5.4}{10} $$ $$ =0.54 $$(ii) By using long division method, we have
\begin{array}{l|l} & 48 \\ \hline 4 & \overline{23} ~ \overline{04} \\ & 16 \\ \hline 88 & ~704 \\ & ~704 \\ \hline & ~~~~~ 0 \\ \end{array}$$ \therefore \quad \sqrt{2304} = 48 $$ $$ \text {And, } \sqrt{.2304} = 0.48 $$\begin{array}{l|l} & 42 \\ \hline 4 & \overline{17} ~ \overline{64} \\ & 16 \\ \hline 82 & ~164 \\ & ~164 \\ \hline & ~~~~~ 0 \\ \end{array}$$ \therefore \quad \sqrt{529} = 42 $$ $$ \text {And, } \sqrt{5.29} = 0.42 $$ $$ \frac{\sqrt{0.2304}+\sqrt{0.1764}}{\sqrt{0.2304}-\sqrt{0.1764}} $$ $$ =\frac{0.48+0.42}{0.48-0.42} $$ $$ =\frac{0.9}{0.06} $$ $$ =15 $$$$ \text {Q17. Evaluate } \sqrt{50625} \text { and hence find the value of } \sqrt{506.25}+\sqrt{5.0625} $$Sol. (i) By using long division method, we have
\begin{array}{l|l} & 225 \\ \hline 2 & 5 ~ \overline{06} ~ \overline{25} \\ & 4 \\ \hline 42 & ~106 \\ & ~84 \\ \hline 445 & ~~~2225 \\ & ~~~2225 \\ \hline & ~~~~~ 0 \\ \end{array}$$ \therefore \quad \sqrt{50625} = 225 $$ $$ \Rightarrow \quad \sqrt{506.25} = \sqrt{\frac{50625}{100}} = \frac {225}{10} = 22.5 $$$$ \Rightarrow \quad \sqrt{5.0625} = \sqrt{\frac{50625}{10000}} = \frac {225}{100} = 2.25 $$Hence,
$$ \sqrt{506.25}+\sqrt{5.0625} = 22.5 + 2.25 = 24.75 $$$$ \text {Q18. Find the value of } \sqrt{103.0225} \text { and hence find the value of } $$ $$ (i) \quad \sqrt{10302.25} $$ $$ (ii) \quad \sqrt{1.030225} $$Sol.
\begin{array}{l|l} & 1015 \\ \hline 1 & 1 ~ \overline{03} ~ \overline{02} ~ \overline{25} \\ & 1 \\ \hline 20 & ~03 \\ & ~0 \\ \hline 201 & ~~~302 \\ & ~~~201 \\ \hline 2025 & ~~~~~10125 \\ & ~~~~~10125 \\ \hline & ~~~~~~ 0 \\ \end{array}$$ \therefore \quad \sqrt{1030225} = 1015 $$ $$ \Rightarrow \quad \sqrt{103.0225} =\sqrt{\frac{1030225}{10000}}= \frac{1015}{100} = 10.15 $$$$ (i) \quad \sqrt{10302.25} $$ $$ \Rightarrow \quad \sqrt{10302.25} = \sqrt{\frac{1030225}{100}} = \frac {1015}{10} = 101.5 $$$$ (ii) \quad \sqrt{1.030225} $$ $$ \Rightarrow \quad \sqrt{1.030225} = \sqrt{\frac{1030225}{1000000}} = \frac {1015}{1000} = 1.015 $$