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

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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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}

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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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

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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Question 4: Find the square root of:

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

Answer:

i) \frac{1089}{4624}

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

ii) 25 \frac{544}{729}

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

iii) 3 \frac{942}{2209}

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

iv) 42 \frac{583}{1369}

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

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Question 5: Find the value of:

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

Answer:

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

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

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

= \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}

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Question 6: Find the value of \sqrt{\frac{3}{8}} correct to 2 places of decimal.

Answer:

\sqrt{ \frac{3}{8}} = \sqrt{\frac{6}{16}} = \frac{\sqrt{6}}{4} = \frac{2.44}{4} =0.61

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

Answer:

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

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

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

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

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

i) 10302.25       ii) 0.01030225

Answer:

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

i) 10302.25

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

ii) 0.01030225

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

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Question 9: Evaluate

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

Answer:

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

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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 }}}}}

Answer:

i)  \sqrt{ \frac{0.289 }{ 0.00121} }

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

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

= \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.

Answer:

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

x^2 = 210.25 or x = 14.5

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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.

Answer:

x^2=218.79 \times 24.13 or x = 72.93

Therefore Perimeter =4 \times 72.93=291.72 meters