Question 1: Find the square root of:

i) $5.4289$     ii) $176.252176$     iii) $18.1476$    iv) $0.018769$     v) $67.0761$     vi) $0.00038809$

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) $0.00038809$

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

$\\$

Question 2:

Question 2: Find the value of each of the following up to 3 decimal places :

i) $\sqrt{2}$      ii) $\sqrt{7}$      iii) $\sqrt{27}$       iv) $\sqrt{2.469}$      v) $\sqrt{25.72}$      vi) $\sqrt{0.4}$

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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Question 3: Find the value of each of the following correct to 2 places of decimal:

i) $\sqrt{11} =3.32$      ii) $\sqrt{2.4} =1.55$     iii) $\sqrt{0.019} = 0.14$       iv) $\sqrt{0.121}= 0.35$

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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Question 4: Find the square root of:

i) $\displaystyle \frac{1089}{4624}$            ii) $\displaystyle 25 \frac{544}{729}$             iii) $\displaystyle 3 \frac{942}{2209}$            iv) $\displaystyle 42 \frac{583}{1369}$

i) $\displaystyle \frac{1089}{4624}$

$\displaystyle \sqrt{\frac{1089}{4624}} = \sqrt{ \frac{33 \times 33}{68 \times 68}} = \frac{33}{68}$

ii) $\displaystyle 25 \frac{544}{729}$

$\displaystyle \sqrt{25 \frac{544}{729} } = \sqrt{\frac{18769}{729}} = \frac{137}{27} =5 \frac{2}{27}$

iii) $\displaystyle 3 \frac{942}{2209}$

$\displaystyle \sqrt{3 \frac{942}{2209}} = \sqrt{\frac{7569}{2209}} = \frac{87}{47} =1 \frac{40}{47}$

iv) $\displaystyle 42 \frac{583}{1369}$

$\displaystyle \sqrt{42 \frac{583}{1369}} = \sqrt{\frac{58081}{1369}} = \frac{241}{37} =6 \frac{19}{37}$

$\displaystyle \\$

Question 5: Find the value of:

i) $\displaystyle \frac{\sqrt{243}}{\sqrt{867}}$       ii) $\displaystyle \frac{\sqrt{4352}}{\sqrt{12393}}$

i) $\displaystyle \frac{\sqrt{243}}{\sqrt{867}}$

$\displaystyle = \sqrt{\frac{243}{867}} = \sqrt{\frac{81}{289}} = \sqrt{ \frac{9 \times 9}{17 \times 17}} = \frac{9}{17}$

ii) $\displaystyle \frac{\sqrt{4352}}{\sqrt{12393}}$

$\displaystyle = \sqrt{\frac{4352}{12393}} = \sqrt{\frac{8 \times 2 \times 2 \times 2 \times 2 \times 2 \times 17 }{3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 17}}= \sqrt{\frac{2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2}{3 \times 3 \times 3 \times 3 \times 3 \times 3}} = \frac{16}{27}$

$\displaystyle \\$

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

$\displaystyle \sqrt{15625} = \sqrt{25 \times 625} = 5 \times 25=125$

$\displaystyle \sqrt{156.25}= \frac{\sqrt{15625}}{\sqrt{100}} = \frac{125}{10} = 12.5$

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

Therefore $\displaystyle \sqrt{156.25} + \sqrt{1.5625} = 13.75$

$\displaystyle \\$

Question 8: Find the value of $\displaystyle \sqrt{103.0225}$ and hence find the square root of:

i) $\displaystyle 10302.25$      ii) $\displaystyle 0.01030225$

$\displaystyle \sqrt{103.0225} = \frac{\sqrt{1030225}}{\sqrt{10000}} = \frac{1015}{100} = 10.15$

i) $\displaystyle 10302.25$

$\displaystyle \sqrt{10302.25} = \frac{\sqrt{1030225}}{\sqrt{100}} = \frac{1015}{10} = 101.5$

ii) $\displaystyle 0.01030225$

$\displaystyle \sqrt{0.01030225} = \frac{\sqrt{1030225}}{\sqrt{100000000}} = \frac{1015}{10000} =0.1015$

$\displaystyle \\$

Question 9: Evaluate

i) $\sqrt{99} \times \sqrt{396}$           ii) $\sqrt{147} \times \sqrt{243}$

i) $\sqrt{99} \times \sqrt{396}$

$=\sqrt{(100-1)} \times \sqrt{(400-4)}$

$=\sqrt{(100-1)} \times \sqrt{(100-1) \times 4}$

$=(100-1) \times 2=99 \times 2=198$

ii) $\sqrt{147} \times \sqrt{243}$

$=\sqrt{7 \times 7 \times 3} \times \sqrt{3 \times 9 \times 9}$

$=7 \times 3 \times 9=189$

$\\$

Question 10: Evaluate

i)  $\sqrt{ \frac{0.289 }{ 0.00121} }$     ii) $\sqrt{ \frac{48.4 }{ 0.289}}$     iii) $\sqrt{ 0.01+\sqrt{0.0064}}$

iv) $\sqrt{0.01} + \sqrt{0.81} + \sqrt{1.21} + \sqrt{0.0009}$     v) $\sqrt{ 41-\sqrt{ 21+\sqrt{ 19- \sqrt{9}}}}$

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

i)  $\sqrt{ \frac{0.289 }{ 0.00121} }$

$\displaystyle = \sqrt{ \frac{0.289 }{ 0.00121}} = \sqrt{ \frac{28900 }{ 121}} \frac{170 }{ 11} =15 \frac{5 }{ 11}$

ii) $\sqrt{ \frac{48.4 }{ 0.289}}$

$\displaystyle = \sqrt{ \frac{48.4 }{ 0.289}} = \sqrt{ \frac{48400 }{ 289}} = \frac{220 }{ 17} =12 \frac{16 }{ 17}$

iii) $\sqrt{ 0.01+\sqrt{0.0064}}$

$=\sqrt{ 0.01+\sqrt{0.0064}}$

$=\sqrt{ 0.01+0.08}$

$=\sqrt{0.09} =0.03$

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

$=0.1+0.09+1.1+0.03=2.13$

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

$=\sqrt{ 41-\sqrt{ 21+\sqrt{ 19- 3}}}$

$=\sqrt{ 41-\sqrt{ 21+4 }}$

$=\sqrt{ 41-5} =6$

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

$=\sqrt{ 10+\sqrt{ 25+\sqrt{ 108+ \sqrt{ 154+15 }}}}$

$=\sqrt{ 10+\sqrt{ 25+\sqrt{ 108+ 13 }}}$

$=\sqrt{ 10+\sqrt{ 25+11 }}$

$=\sqrt{ 10+6} =4$

Question 11: Three-fifth of the square of a certain number is $126.15$. Find the number.

$\frac{3}{5}$ $x^2 = 126.15$

$x^2 = 210.25$ or $x = 14.5$

$\\$

Question 12: A square plot of land has area equal to that of a rectangular plot having length $218.79$ meters and breadth $24.31$ meters. Find the perimeter of the square plot.

$x^2=218.79 \times 24.13$ or $x = 72.93$
Therefore Perimeter $=4 \times 72.93=291.72$ meters