\[ \begin{array}{c|c} 3 & 225 \\ \hline 3 & 75 \\ \hline 5 & 25 \\ \hline 5 & 5 \\ \hline & 1 \\ \end{array} \] \[ \begin{align*} \sqrt{225} &= \sqrt{ \boxed{3 \times 3} \times \boxed{5 \times 5}} \\ &= 3 \times 5 \\ &= 15 \\ \end{align*} \]

Answer \( \color{red}\sqrt{225} = 15 \)

\[ \begin{array}{c|c} 3 & 441 \\ \hline 3 & 147 \\ \hline 7 & 49 \\ \hline 7 & 7 \\ \hline & 1 \\ \end{array} \] \[ \begin{align*} \sqrt{441} &= \sqrt{ \boxed{3 \times 3} \times \boxed{7 \times 7}} \\ &= 3 \times 7 \\ &= 21 \\ \end{align*} \]

Answer \( \color{red}\sqrt{441} = 21 \)

\[ \begin{array}{c|c} 23 & 529 \\ \hline 23 & 23 \\ \hline & 1 \\ \end{array} \] \[ \begin{align*} \sqrt{529} &= \sqrt{ \boxed{23 \times 23}} \\ &= 23 \\ \end{align*} \]

Answer \( \color{red}\sqrt{529} = 23 \)

\[ \begin{array}{c|c} 2 & 40000 \\ \hline 2 & 20000 \\ \hline 2 & 10000 \\ \hline 2 & 5000 \\ \hline 2 & 2500 \\ \hline 2 & 1250 \\ \hline 5 & 625 \\ \hline 5 & 125 \\ \hline 5 & 25 \\ \hline 5 & 5 \\ \hline & 1 \\ \end{array} \] \[ \begin{align*} \sqrt{40000} &= \sqrt{\boxed{2 \times 2 } \times \boxed{2 \times 2} \times \boxed{2 \times 2} \times \boxed{5 \times 5} \times \boxed{5 \times 5}} \\ &= 2 \times 2 \times 2 \times 5 \times 5\\ &= 8 \times 25 \\ &= 200 \\ \end{align*} \]

Answer \( \color{red}\sqrt{40000} = 200 \)

\[ \begin{array}{c|c} 2 & 7744 \\ \hline 2 & 3872 \\ \hline 2 & 1936 \\ \hline 2 & 968 \\ \hline 2 & 484 \\ \hline 2 & 242 \\ \hline 11 & 121 \\ \hline 11 & 11 \\ \hline & 1 \\ \end{array} \] \[ \begin{align*} \sqrt{7744} &= \sqrt{\boxed{2 \times 2} \times \boxed{2 \times 2} \times \boxed{2 \times 2} \times \boxed{11 \times 11}} \\ &= 2 \times 2 \times 2 \times 11 \\ &= 8 \times 11 \\ &= 88 \\ \end{align*} \]

Answer \( \color{red}\sqrt{7744} = 88 \)

\[ \begin{array}{c|c} 7 & 8281 \\ \hline 7 & 1183 \\ \hline 13 & 169 \\ \hline 13 & 13 \\ \hline & 1 \\ \end{array} \] \[ \begin{align*} \sqrt{8281} &= \sqrt{\boxed{7 \times 7} \times \boxed{13 \times 13}} \\ &= 7 \times 13 \\ &= 7 \times 13 \\ &= 91 \\ \end{align*} \]

Answer \( \color{red}\sqrt{8281} = 91 \)

\[ \begin{array}{c|c} 2 & 4096 \\ \hline 2 & 2048 \\ \hline 2 & 1024 \\ \hline 2 & 512 \\ \hline 2 & 256 \\ \hline 2 & 128 \\ \hline 2 & 64 \\ \hline 2 & 32 \\ \hline 2 & 16 \\ \hline 2 & 8 \\ \hline 2 & 4 \\ \hline 2 & 2 \\ \hline & 1 \\ \end{array} \] \[ \begin{align*} \sqrt{4096} &= \sqrt{\boxed{2 \times 2} \times \boxed{2 \times 2} \times \boxed{2 \times 2} \times \boxed{2 \times 2} \times \boxed{2 \times 2} \times \boxed{2 \times 2}} \\ &= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \\ &= 64 \\ \end{align*} \]

Answer \( \color{red}\sqrt{4096} = 64 \)

\[ \begin{array}{c|c} 2 & 28900 \\ \hline 2 & 14450 \\ \hline 5 & 7225 \\ \hline 5 & 1445 \\ \hline 17 & 289 \\ \hline 17 & 17 \\ \hline & 1 \\ \end{array} \] \[ \begin{align*} \sqrt{28900} &= \sqrt{\boxed{2 \times 2} \times \boxed{5 \times 5} \times \boxed{17 \times 17}} \\ &= 2 \times 5 \times 17 \\ &= 10 \times 17 \\ &= 170 \\ \end{align*} \]

Answer \( \color{red}\sqrt{28900} = 170 \)

\[ \begin{array}{c|c} 2 & 1100 \\ \hline 2 & 550 \\ \hline 5 & 275 \\ \hline 5 & 55 \\ \hline 11 & 11 \\ \hline & 1 \\ \end{array} \] \[ \begin{align*} 1100 &= \boxed{2 \times 2} \times \boxed{5 \times 5} \times 11 \\ \\ & \text{Required number is } 11 \\ & \text{Multiply by } \color{red}\boxed{11} \\ 1100 \times 11 &= 12100 \\ \sqrt{12100} & = \sqrt{\boxed{2 \times 2} \times \boxed{5 \times 5} \times \boxed{11 \times 11}} \\ & = 2 \times 5 \times 11 \\ \sqrt{12100} & = \color{red}\boxed{110} \\ \end{align*} \]

Answer The smallest number to multiply by 1100 to get a perfect square is \( \color{red}\boxed{11} \), and the square root of the perfect square 12100 is \( \color{red}\boxed{110} \).

\[ \begin{array}{c|c} 2 & 180 \\ \hline 2 & 90 \\ \hline 3 & 45 \\ \hline 3 & 15 \\ \hline 5 & 5 \\ \hline & 1 \\ \end{array} \] \[ \begin{align*} 180 &= \boxed{2 \times 2} \times \boxed{3 \times 3} \times 5 \\ \\ & \text{Multiply by } \color{red}\boxed{5} \\ 180 \times 5 &= 900 \\ \sqrt{900} & = \sqrt{\boxed{2 \times 2} \times \boxed{3 \times 3} \times \boxed{5 \times 5}} \\ & = 2 \times 3 \times 5 \\ \sqrt{900} & = \color{red}\boxed{30} \\ \end{align*} \]

Answer The smallest number to multiply by 180 to get a perfect square is \( \color{red}\boxed{5} \), and the square root of the perfect square 900 is \( \color{red}\boxed{30} \).

\[ \begin{array}{c|c} 3 & 3645 \\ \hline 3 & 1215 \\ \hline 3 & 405 \\ \hline 3 & 135 \\ \hline 3 & 45 \\ \hline 5 & 15 \\ \hline 3 & 5 \\ \hline & 1 \\ \end{array} \] \[ \begin{align*} 3645 &= \boxed{3 \times 3} \times \boxed{3 \times 3} \times \boxed{3 \times 3} \times 5 \\ \\ & \text{Divide by } \color{red}\boxed{5} \\ \frac{3645}{5} &= 729 \\ \sqrt{729} & = \sqrt{\boxed{3 \times 3} \times \boxed{3 \times 3} \times \boxed{3 \times 3}} \\ & = 3 \times 3 \times 3 \\ & = \color{red}\boxed{27} \\ \end{align*} \]

Answer The smallest number by which 3645 must be divided to get a perfect square is \( \color{red}\boxed{5} \), and the square root of the resulting number 729 is \( \color{red}\boxed{27} \).

\[ \begin{align*} \text{No. of rows} &= x \\ \text{No. of trees in each row} &= x \\ \text{Total no. of trees} &= 1521 \\ x \times x &= 1521 \\ x^2 &= 1521 \\ x &= \sqrt{1521} \\ \end{align*} \] \[ \begin{array}{c|c} 3 & 1521 \\ \hline 3 & 507 \\ \hline 3 & 169 \\ \hline 13 & 169 \\ \hline 13 & 13 \\ \hline & 1 \\ \end{array} \] \[ \begin{align*} \sqrt{1521} &= \sqrt{\boxed{3 \times 3} \times \boxed{13 \times 13}} \\ &= 3 \times 13 \\ &= 39 \\ \end{align*} \]

Answer Number of rows \( \color{red}\boxed{39} \).

\[ \begin{align*} \text{No. of rows} &= x \\ \text{No. of cadets in each row} &= x \\ \text{Total no. of cadets} &= 202500 \\ x \times x &= 202500 \\ x^2 &= 202500 \\ x &= \sqrt{202500} \\ \end{align*} \] \[ \begin{array}{c|c} 2 & 202500 \\ \hline 2 & 101250 \\ \hline 3 & 50625 \\ \hline 3 & 16875 \\ \hline 3 & 5625 \\ \hline 3 & 1875 \\ \hline 5 & 625 \\ \hline 5 & 125 \\ \hline 5 & 25 \\ \hline 5 & 5 \\ \hline & 1 \\ \end{array} \] \[ \begin{align*} \sqrt{202500} &= \sqrt{\boxed{2 \times 2} \times \boxed{3 \times 3} \times \boxed{3 \times 3} \times \boxed{5 \times 5} \times \boxed{5 \times 5}} \\ &= 2 \times 3 \times 3 \times 5 \times 5 \\ &= 450 \\ \end{align*} \]

Answer Number of cadets in each row is \( \color{red}\boxed{450} \).

\[ \begin{align*} \text{Area of the square field} &= 5184 \text{ m}^2 \\ \text{Side} \times \text{Side} &= 5184 \\ \text{(Side)}^2 &= 5184 \\ \text{Side} &= \sqrt{5184} \\ \begin{array}{c|c} 2 & 5184 \\ \hline 2 & 2592 \\ \hline 2 & 1296 \\ \hline 2 & 648 \\ \hline 2 & 324 \\ \hline 2 & 162 \\ \hline 3 & 81 \\ \hline 3 & 27 \\ \hline 3 & 9 \\ \hline 3 & 3 \\ \hline & 1 \\ \end{array} \end{align*} \] \[ \begin{align*} \text{Side} &= \sqrt{\boxed{2 \times 2} \times \boxed{2 \times 2} \times \boxed{2 \times 2} \times \boxed{3 \times 3} \times \boxed{3 \times 3}} \\ \text{Side} &= 2 \times 2 \times 2 \times 3 \times 3 \\ \text{Side} &= 72 \text{ m} \\ \\ \end{align*} \] \[ \begin{align*} \text{Perimeter of the square field} &= 4 \times \text{Side} \\ &= 4 \times 72 \\ &= 288 \text{ m}\\ \\ \text{Let breadth of the rectangular field} &= x \\ \text{Let length of the rectangular field} &= 2x \\ \text{Perimeter of the rectangular field} &= 288 \text{ m} \\ 2\text{ (l + b)} &= 288 \text{ m} \\ 2 (x + 2x) &= 288 \text{ m} \\ 2 (3x) &= 288 \text{ m} \\ 6x &= 288 \text{ m} \\ x &= \frac{288}{6} \text{ m} \\ x &= 48 \text{ m} \\ \text{Length} &= 96 \text{ m}\\ \text{Breadth} &= 48 \text{ m}\\ \\ \text{Area of the rectangular field} &= \text{ l} \times \text{ b} \\ &= 96 \times 48 \\ &= 4608 \text{ m}^2 \\ \end{align*} \]

Answer Area of the rectangular field \( \color{red}\boxed{4608 \text{ m}^2} \)

\[ \begin{array}{c|c} 7 & 47089 \\ \hline 7 & 6727 \\ \hline 31 & 961 \\ \hline 31 & 31 \\ \hline & 1 \\ \end{array} \quad \quad \begin{array}{c|c} 2 & 24336 \\ \hline 2 & 12168 \\ \hline 2 & 6084 \\ \hline 2 & 3042 \\ \hline 3 & 1521 \\ \hline 3 & 507 \\ \hline 13 & 169 \\ \hline 13 & 13 \\ \hline & 1 \\ \end{array} \] \[ \begin{align*} \sqrt{47089} &= \sqrt{\boxed{7 \times 7} \times \boxed{31 \times 31}} \\ &= 7 \times 31 \\ \color{green} \sqrt{47089} &= \color{green} 217 \\ \\ \sqrt{24336} &= \sqrt{\boxed{2 \times 2} \times \boxed{2 \times 2} \times \boxed{3 \times 3} \times \boxed{13 \times 13}} \\ &= 2 \times 2 \times 3 \times 13 \\ \color{green} \sqrt{24336} &= \color{green} 156 \\ \\ \sqrt{47089} + \sqrt{24336} &= 217 + 156 \\ &= 373 \\ \end{align*} \]

Answer \( \sqrt{47089} + \sqrt{24336} \) = \( \color{red}\boxed{373} \)