10. Given the prime factorisation of each of the following numbers, find its square root without using a calculator. \( \begin{array}{ll}\text { (a) } 9801=3^{4} \times 11^{2} & \text { (b) } 35721=3^{6} \times 7^{2}\end{array} \)

To find the square roots of the given numbers using their prime factorizations, follow these steps: ### (a) \(9801 = 3^{4} \times 11^{2}\) 1. **Understand the Square Root of a Product:** \[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \] 2. **Apply the Square Root to Each Prime Factor:** \[ \sqrt{3^{4} \times 11^{2}} = \sqrt{3^{4}} \times \sqrt{11^{2}} \] 3. **Simplify Each Square Root:** \[ \sqrt{3^{4}} = 3^{2} = 9 \] \[ \sqrt{11^{2}} = 11 \] 4. **Multiply the Results:** \[ 9 \times 11 = 99 \] **So, \(\sqrt{9801} = 99\).** --- ### (b) \(35721 = 3^{6} \times 7^{2}\) 1. **Understand the Square Root of a Product:** \[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \] 2. **Apply the Square Root to Each Prime Factor:** \[ \sqrt{3^{6} \times 7^{2}} = \sqrt{3^{6}} \times \sqrt{7^{2}} \] 3. **Simplify Each Square Root:** \[ \sqrt{3^{6}} = 3^{3} = 27 \] \[ \sqrt{7^{2}} = 7 \] 4. **Multiply the Results:** \[ 27 \times 7 = 189 \] **So, \(\sqrt{35721} = 189\).** --- **Final Answers:** - **(a)** \(\sqrt{9801} = 99\) - **(b)** \(\sqrt{35721} = 189\)