Solution

\[ \begin{align*} \text{Total cost of levelling} & = \text{₹}19935\\ \text{Cost of levelling per square metre} & = \text{₹}15\\ \text{Area of Square lawn} & = \frac{19935}{15}\\ \color{green} \text{Area of Square lawn} & = \color{green} 1329 \ m^2 \\ ( Side)^2 & = 1329 \ m^2 \\ Side & = \sqrt{1329} \\ \end{align*} \] \[ \begin{array}{r|l} & \phantom{0}36.45 \\ \hline 3 & \phantom{-}\overline{13} \, \overline{29} . \overline{00} \, \overline{00} \\ & -\phantom{0}9 \\ \hline 66 & \phantom{000}429 \\ & \ \phantom{ }-396 \\ \hline 724 & \phantom{0000}3300 \\ & \phantom{-}-2896 \\ \hline 7285& \phantom{00000}40400 \\ & \ \phantom{00}-36425 \\ \hline & \ \phantom{---}3975 \\ \end{array} \] \[ \begin{align*} \sqrt{1329} &= 36.45 \\ Side & = 36.45 \\ \color{green} \text{Side of Square lawn} & \approx \color{green} 36.5m \\ \\ \text{Perimeter of Square lawn} & = 4 \times Side \\ & = 4 \times 36.5 \\ & = 146m \\ \\ \text{Cost of fencing per metre} & = \text{₹} 22 \\ \text{Total cost of fencing} & = 22 \times 146 \\ & = \color{green} \text{₹} 3212 \end{align*} \]

Answer Total cost of fencing \( = \color{red} \text{₹} 3212 \)

Solution

\[ \begin{array}{c|c} 2 & 72 \\ \hline 2 & 36 \\ \hline 2 & 18 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} \quad \quad \begin{array}{c|c} 2 & 48 \\ \hline 2 & 24 \\ \hline 2 & 12 \\ \hline 2 & 6 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} \] \[ \begin{align*} & = \sqrt{72} + \sqrt{48} \\ & = \sqrt{2 \times 2 \times 2 \times 3 \times 3} + \sqrt{2 \times 2 \times 2 \times 2 \times 3} \\ & = ( 2 \times 3 \times \sqrt{2} )+ ( 2 \times 2 \times \sqrt{3} ) \\ & = ( 6 \times \sqrt{2} )+ ( 4 \times \sqrt{3} ) \\ & = ( 6 \times 1.414 )+ ( 4 \times 1.732 ) \\ & = 8.484 + 6.928 \\ & = 15.412 \end{align*} \]

Answer \( \sqrt{72} + \sqrt{48} = \color{red} 15.412 \)

Solution

\[ \begin{array}{c|c} 5 & 125 \\ \hline 5 & 25 \\ \hline 5 & 5 \\ \hline & 1 \\ \end{array} \quad \quad \begin{array}{c|c} 2 & 64 \\ \hline 2 & 32 \\ \hline 2 & 16 \\ \hline 2 & 8 \\ \hline 2 & 4 \\ \hline 2 & 2 \\ \hline & 1 \\ \end{array} \] \[ \begin{align*} & = \sqrt{\frac{125}{64}} \\ \\ & = \sqrt{\frac{5 \times 5 \times 5}{2 \times 2 \times 2 \times 2 \times 2 \times 2}} \\ \\ & = \frac{5 \times \sqrt 5}{2 \times 2 \times 2} \\ \\ & = \frac{5 \times 2.236}{8} \\ \\ & = \frac{11.18}{8} \\ \\ & = 1.3975 \end{align*} \]

Answer \(\displaystyle \sqrt{\frac{125}{64}} = \color{red} 1.3975 \)