Class 8: Square Roots – Exercise 7B

$\displaystyle \textbf{Question 1: }\text{Find the square root of:}$
$\displaystyle \text{(i) }5.4289\qquad\text{(ii) }176.252176\qquad \text{(iii) }18.1476$
$\displaystyle \text{(iv) }0.018769\qquad\text{(v) }67.0761\qquad \text{(vi) }0.00038809$
$\displaystyle \text{Answer:}$

i) $5.4289$

$\begin{array}{r | r r r r r} 2 & \overline{5} & . & \overline{42} & \overline{89} & ( 2.33 \\ & 4 & & & & \\ \hline 43 & 1 & & 42 & & \\ & 1 & & 29 & & \\ \hline 463 & & & 13 & 89 & \\ & & & 13 & 89 & \\ \hline & & & 0 & & \end{array}$

ii) $176.252176$

$\begin{array}{r | r r r r r r r } 1 & \overline{1} & \overline{76} & . & \overline{25} & \overline{21} & \overline{76} & ( 13.276 \\ & 1 & & & & & & \\ \hline 23 & & 76 & & & & & \\ & & 69 & & & & & \\ \hline 262 & & 7 & & 25 & & & \\ & & 5 & & 24 & & & \\ \hline 2647 & & 2 & & 01 & 21 & & \\ & & 1 & & 85 & 29 & & \\ \hline 26546 & & & & 15 & 92 & 76 & \\ & & & & 15 &92 & 76 & \\ \hline & & & & & 0 & & \end{array}$

iii) $18.1476$

$\begin{array}{r | r r r r r } 4 & \overline{18} & . & \overline{14} & \overline{76} & (4.26 \\ & 16 & & & & \\ \hline 82 & 2 & & 14 & & \\ & 1 & & 64 & & \\ \hline 846 & & & 50 & 76 \\ & & & 50 & 76 \\ \hline & & & 0 &\end{array}$

iv) $0.018769$

$\begin{array}{r | r r r r r r } 1 & \overline{0} & . & \overline{01} & \overline{87} & \overline{69} & ( 0.137 \\ & & & 1 & & & \\ \hline 23 & & & 0 & 87 & & \\ & & & & 69 & & \\ \hline 262 & & & & 18 & 69 & \\ & & & & 18 & 69 & \\ \hline & & & & & 0 & \\ \end{array}$

v) $67.0761$

$\begin{array}{r | r r r r r } 8 & \overline{67} & . & \overline{07} & \overline{61} & ( 8.19 \\ & 64 & & & & \\ \hline 161 & 3 & & 07 & & \\ & 1 & & 61 & & \\ \hline 1629 & 1 & & 46 & 61 & \\ & 1 & & 46 & 61 & \\ \hline & & & & 0 & \end{array}$

vi) $0.00038809$

$\begin{array}{r | r r r r r r r } 1 & \overline{0} & . & \overline{00} & \overline{03} & \overline{88} & \overline{09} & ( 0.0197 \\ & & & 01 & & & \\ \hline 29 & & & & 2 & 88 & & \\ & & & & 2 & 61 & & \\ \hline 387 & & & & & 27 & 09 & \\ & & & & & 27 & 09 & \\ \hline & & & & & 0 & & \end{array}$

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$\displaystyle \textbf{Question 2: }\text{Find the value of each of the following up to 3 decimal}$
$\displaystyle \text{places:}$
$\displaystyle \text{(i) }\sqrt{2}\qquad\text{(ii) }\sqrt{7}\qquad \text{(iii) }\sqrt{27}$
$\displaystyle \text{(iv) }\sqrt{2.469}\qquad\text{(v) }\sqrt{25.72}\qquad \text{(vi) }\sqrt{0.4}$
$\displaystyle \text{Answer:}$

i) $\sqrt{2}$

$\begin{array}{r | r r r r r r } 1 & \overline{2} & . & \overline{00} & \overline{00} & \overline{00} & ( 1.414 \\ & 1 & & & & & \\ \hline 24 & 1 & & 00 & & & \\ & & & & 96 & & \\ \hline 281 & & & 4 & 00 & & \\ & & & 2 & 81 & & \\ \hline 2824 & & & 1 & 19 & 00 & \\ \hline & & & 1 & 12 & 96 & \\ \hline & & & 6 & 14 & 14 & \end{array}$

ii) $\sqrt{7}$

$\begin{array}{r | r r r r r r } 2 & \overline{7} & . & \overline{00} & \overline{00} & \overline{00} & ( 2.645 \\ & 4 & & & & & \\ \hline 46 & 3 & & 00 & & & \\ & 2 & & 76 & & & \\ \hline 524 & & & 24 & 00 & & \\ & & & 20 & 96 & & \\ \hline 5285 & & & 3 & 14 & 00 & \\ \hline & & & 2 & 64 & 25 & \\ \hline & & & & 49 & 75 & \end{array}$

iii) $\sqrt{27}$

$\begin{array}{r | r r r r r r } 5 & \overline{27} & . & \overline{00} & \overline{00} & \overline{00} & ( 5.196 \\ & 25 & & & & & \\ \hline 101 & 2 & & 00 & & & \\ & 1 & & 01 & & & \\ \hline 1029 & & & 99 & 00 & & \\ & & & 92 & 61 & & \\ \hline 10386 & & & 6 & 39 & 00 & \\ \hline & & & 6 & 23 & 16 & \\ \hline & & & & 15 & 74 & \end{array}$

iv) $\sqrt{2.469}$

$\begin{array}{r | r r r r r r } 1 & \overline{2} & . & \overline{46} & \overline{90} & \overline{00} & ( 1.571 \\ & 1 & & & & & \\ \hline 25 & 1 & & 46 & & & \\ & 1 & & 25 & & & \\ \hline 307 & & & 21 & 90 & & \\ & & & 21 & 49 & & \\ \hline 3141 & & & & 41 & 00 & \\ \hline & & & & 31 & 41 & \\ \hline & & & & 9 & 59 & \end{array}$

v) $\sqrt{25.72}$

$\begin{array}{r | r r r r r r } 5 & \overline{25} & . & \overline{72} & \overline{00} & \overline{00} & ( 5.071 \\ & 25 & & & & & \\ \hline 1007 & & & 72 & 00 & & \\ & & & 70 & 49 & & \\ \hline 10141 & & & 1 & 51 & 00 & \\ \hline & & & 1 & 01 & 41 & \\ & & & & 49 & 59 & \end{array}$

vi) $\sqrt{0.4}$

$\begin{array}{r | r r r r r r } 6 & \overline{0} & . & \overline{40} & \overline{00} & \overline{00} & ( 0.632 \\ & & & 36 & & & \\ \hline 123 & & & 4 & 00 & & \\ & & & 3 & 69 & & \\ \hline 1262 & & & & 31 & 00 & \\ \hline & & & & 25 & 24 & \\ & & & & 5 & 72 & \end{array}$

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$\displaystyle \textbf{Question 3: }\text{Find the value of each of the following correct to 2 places}$
$\displaystyle \text{of decimal:}$
$\displaystyle \text{(i) }\sqrt{11}\qquad\text{(ii) }\sqrt{2.4}\qquad \text{(iii) }\sqrt{0.019}$
$\displaystyle \text{(iv) }\sqrt{0.121}$
$\displaystyle \text{Answer:}$

i) $\sqrt{11} =3.32$

$\begin{array}{r | r r r r r r } 3 & \overline{11} & . & \overline{00} & \overline{00} & \overline{00} & (3.316 \\ & 9 & & & & & \\ \hline 63 & 2 & & 00 & & & \\ & 1 & & 89 & & & \\ \hline 661 & & & 11 & 00 & & \\ & & & 6 & 61 & & \\ \hline 6626 & & & 4 & 39 & 00 & \\ & & & 3 & 97 & 56 & \\ \hline & & & & 41 & 44 & \end{array}$

ii) $\sqrt{2.4} =1.55$

$\begin{array}{r | r r r r r r } 1 & \overline{2} & . & \overline{40} & \overline{00} & \overline{00} & (1.549 \\ & 1 & & & & & \\ \hline25 & 1 & & 40 & & & \\ & 1 & & 25 & & & \\ \hline304 & & & 15 & 00 & & \\ & & & 12 & 16 & & \\ \hline3089 & & & 2 & 84 & 00 & \\ & & & 2 & 78 & 01 & \\ \hline & & & & 5 & 99 & \end{array}$

iii) $\sqrt{0.019} = 0.14$

$\begin{array}{r | r r r r r r } 1 & \overline{0} & . & \overline{01} & \overline{90} & \overline{00} & (0.137 \\ & & & 01 & & & \\ \hline & & & & 90 & & \\ & & & & 69 & & \\ \hline 267 & & & & 21 & 00 & \\ & & & & 18 & 69 & \\ \hline & & & & 2 & 31 & \end{array}$

iv) $\sqrt{0.121}= 0.35$

$\begin{array}{r | r r r r r r } 3 & \overline{0} & . & \overline{12} & \overline{10} & \overline{00} & (0.347 \\ & & & 9 & & & \\ \hline 64 & & & 3 & 10 & & \\ & & & 2 & 56 & & \\ \hline 687 & & & & 54 & 00 & \\ & & & & 48 & 09 & \\ \hline & & & & 5 & 91 & \end{array}$

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$\displaystyle \textbf{Question 4: }\text{Find the square root of:}$
$\displaystyle \text{(i) }\frac{1089}{4624}\qquad \text{(ii) }25\frac{544}{729}$

$\displaystyle \text{(iii) }3\frac{942}{2209}\qquad \text{(iv) }42\frac{583}{1369}$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) }\sqrt{\frac{1089}{4624}} =\sqrt{\frac{33\times33}{68\times68}}=\frac{33}{68}.$

$\displaystyle \text{(ii) }\sqrt{25\frac{544}{729}} =\sqrt{\frac{18769}{729}}$
$\displaystyle =\sqrt{\frac{137\times137}{27\times27}} =\frac{137}{27}=5\frac{2}{27}.$

$\displaystyle \text{(iii) }\sqrt{3\frac{942}{2209}} =\sqrt{\frac{7569}{2209}}$
$\displaystyle =\sqrt{\frac{87\times87}{47\times47}} =\frac{87}{47}=1\frac{40}{47}.$

$\displaystyle \text{(iv) }\sqrt{42\frac{583}{1369}} =\sqrt{\frac{58081}{1369}}$
$\displaystyle =\sqrt{\frac{241\times241}{37\times37}} =\frac{241}{37}=6\frac{19}{37}.$
$\displaystyle \\$

$\displaystyle \textbf{Question 5: }\text{Find the value of:}$
$\displaystyle \text{(i) }\frac{\sqrt{243}}{\sqrt{867}}\qquad \text{(ii) }\frac{\sqrt{4352}}{\sqrt{12393}}$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) }\frac{\sqrt{243}}{\sqrt{867}} =\sqrt{\frac{243}{867}}=\sqrt{\frac{81}{289}}$
$\displaystyle =\sqrt{\frac{9\times9}{17\times17}}=\frac{9}{17}.$

$\displaystyle \text{(ii) }\frac{\sqrt{4352}}{\sqrt{12393}} =\sqrt{\frac{4352}{12393}}$
$\displaystyle =\sqrt{\frac{2^8\times17}{3^6\times17}} =\sqrt{\frac{2^8}{3^6}}$
$\displaystyle =\frac{2^4}{3^3}=\frac{16}{27}.$
$\displaystyle \\$

$\displaystyle \textbf{Question 7: }\text{Find the value of }\sqrt{15625}\text{ and hence evaluate}$
$\displaystyle \sqrt{156.25}+\sqrt{1.5625}.$

$\displaystyle \text{Answer:}$
$\displaystyle \sqrt{15625}=\sqrt{25\times625}=5\times25=125.$
$\displaystyle \sqrt{156.25} =\sqrt{\frac{15625}{100}}=\frac{\sqrt{15625}}{\sqrt{100}}$
$\displaystyle =\frac{125}{10}=12.5.$

$\displaystyle \sqrt{1.5625} =\sqrt{\frac{15625}{10000}}=\frac{\sqrt{15625}}{\sqrt{10000}}$
$\displaystyle =\frac{125}{100}=1.25.$

$\displaystyle \therefore \sqrt{156.25}+\sqrt{1.5625} =12.5+1.25=13.75.$
$\displaystyle \\$

$\displaystyle \textbf{Question 8: }\text{Find the value of }\sqrt{103.0225}\text{ and hence find the}$
$\displaystyle \text{square root of:}$
$\displaystyle \text{(i) }10302.25\qquad\text{(ii) }0.01030225$

$\displaystyle \text{Answer:}$
$\displaystyle \sqrt{103.0225} =\sqrt{\frac{1030225}{10000}}=\frac{1015}{100}=10.15.$

$\displaystyle \text{(i) }\sqrt{10302.25} =\sqrt{\frac{1030225}{100}}=\frac{1015}{10}=101.5.$

$\displaystyle \text{(ii) }\sqrt{0.01030225} =\sqrt{\frac{1030225}{100000000}}$
$\displaystyle =\frac{1015}{10000}=0.1015.$
$\displaystyle \\$

$\displaystyle \textbf{Question 9: }\text{Evaluate:}$
$\displaystyle \text{(i) }\sqrt{99}\times\sqrt{396}\qquad \text{(ii) }\sqrt{147}\times\sqrt{243}$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) }\sqrt{99}\times\sqrt{396} =\sqrt{99}\times\sqrt{99\times4}$
$\displaystyle =\sqrt{99}\times2\sqrt{99} =2\times99=198.$

$\displaystyle \text{(ii) }\sqrt{147}\times\sqrt{243} =\sqrt{7\times7\times3}\times\sqrt{3\times9\times9}$
$\displaystyle =7\sqrt{3}\times9\sqrt{3} =7\times9\times3=189.$
$\displaystyle \\$

$\displaystyle \textbf{Question 10: }\text{Evaluate:}$
$\displaystyle \text{(i) }\sqrt{\frac{0.289}{0.00121}}\qquad \text{(ii) }\sqrt{\frac{48.4}{0.289}}$

$\displaystyle \text{(iii) }\sqrt{0.01+\sqrt{0.0064}}$

$\displaystyle \text{(iv) }\sqrt{0.01}+\sqrt{0.81}+\sqrt{1.21} +\sqrt{0.0009}$

$\displaystyle \text{(v) }\sqrt{41-\sqrt{21+\sqrt{19-\sqrt{9}}}}$

$\displaystyle \text{(vi) }\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}}$

$\displaystyle \text{Answer:}$
$\displaystyle \text{(i) }\sqrt{\frac{0.289}{0.00121}} =\sqrt{\frac{28900}{121}}$
$\displaystyle =\frac{170}{11}=15\frac{5}{11}.$

$\displaystyle \text{(ii) }\sqrt{\frac{48.4}{0.289}} =\sqrt{\frac{48400}{289}}$
$\displaystyle =\frac{220}{17}=12\frac{16}{17}.$

$\displaystyle \text{(iii) }\sqrt{0.01+\sqrt{0.0064}} =\sqrt{0.01+0.08}$
$\displaystyle =\sqrt{0.09}=0.3.$

$\displaystyle \text{(iv) }\sqrt{0.01}+\sqrt{0.81}+\sqrt{1.21} +\sqrt{0.0009}$
$\displaystyle =0.1+0.9+1.1+0.03=2.13.$

$\displaystyle \text{(v) }\sqrt{41-\sqrt{21+\sqrt{19-\sqrt{9}}}}$
$\displaystyle =\sqrt{41-\sqrt{21+\sqrt{19-3}}}$
$\displaystyle =\sqrt{41-\sqrt{21+4}}$
$\displaystyle =\sqrt{41-5}=\sqrt{36}=6.$

$\displaystyle \text{(vi) }\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}}$
$\displaystyle =\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+15}}}}$
$\displaystyle =\sqrt{10+\sqrt{25+\sqrt{108+13}}}$
$\displaystyle =\sqrt{10+\sqrt{25+11}}$
$\displaystyle =\sqrt{10+6}=\sqrt{16}=4.$
$\displaystyle \\$

$\displaystyle \textbf{Question 11: }\text{Three-fifth of the square of a certain number is }126.15.$
$\displaystyle \text{Find the number.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let the number be }x.$
$\displaystyle \frac{3}{5}x^2=126.15.$
$\displaystyle x^2=126.15\times\frac{5}{3}=210.25.$
$\displaystyle x=\pm\sqrt{210.25}=\pm14.5.$
$\displaystyle \therefore \text{The numbers are }14.5\text{ and }-14.5.$
$\displaystyle \\$

$\displaystyle \textbf{Question 12: }\text{A square plot of land has area equal to that of a}$
$\displaystyle \text{rectangular plot having length }218.79\text{ metres and breadth }24.31$
$\displaystyle \text{metres. Find the perimeter of the square plot.}$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let the side of the square plot be }x\text{ metres}.$
$\displaystyle x^2=218.79\times24.31.$
$\displaystyle x^2=5318.7849.$
$\displaystyle x=\sqrt{5318.7849}=72.93\text{ metres}.$
$\displaystyle \text{Perimeter of the square}=4x=4\times72.93=291.72\text{ metres}.$
$\displaystyle \therefore \text{The perimeter of the square plot is }291.72\text{ metres}.$
$\displaystyle \\$