Simplify the following without using a calculator: \( \begin{array}{lll}\text { (a) } \sqrt{7} \sqrt{3} & \text { (b) } \sqrt{7} \sqrt{7}+(\sqrt{11})^{2} & \text { (o) }(2 \sqrt{3})^{2} \\ \text { (d) } 3 \sqrt{6}-\sqrt{6}+7 \sqrt{6} & \text { (c) } \sqrt[4]{3}+7 \sqrt[4]{3}-5 \sqrt[4]{3} & \text { (0) } \sqrt{3}+\sqrt{27} \\ \text { (g) } 2 \sqrt{18}-\sqrt{32} & \text { (h) } \frac{\sqrt{32}}{\sqrt{2}} & \text { (i) } \frac{\sqrt{51}}{\sqrt{3}}\end{array} \)

Sure, let's simplify each expression step by step without using a calculator. ### (a) \( \sqrt{7} \times \sqrt{3} \) \[ \sqrt{7} \times \sqrt{3} = \sqrt{7 \times 3} = \sqrt{21} \] **Simplified Form:** \( \sqrt{21} \) --- ### (b) \( \sqrt{7} \times \sqrt{7} + (\sqrt{11})^{2} \) \[ \sqrt{7} \times \sqrt{7} = 7 \] \[ (\sqrt{11})^2 = 11 \] \[ 7 + 11 = 18 \] **Simplified Form:** \( 18 \) --- ### (o) \( (2 \sqrt{3})^{2} \) \[ (2 \sqrt{3})^{2} = 2^2 \times (\sqrt{3})^2 = 4 \times 3 = 12 \] **Simplified Form:** \( 12 \) --- ### (d) \( 3 \sqrt{6} - \sqrt{6} + 7 \sqrt{6} \) Combine like terms: \[ 3 \sqrt{6} - \sqrt{6} + 7 \sqrt{6} = (3 - 1 + 7) \sqrt{6} = 9 \sqrt{6} \] **Simplified Form:** \( 9 \sqrt{6} \) --- ### (c) \( \sqrt[4]{3} + 7 \sqrt[4]{3} - 5 \sqrt[4]{3} \) Combine like terms: \[ \sqrt[4]{3} + 7 \sqrt[4]{3} - 5 \sqrt[4]{3} = (1 + 7 - 5) \sqrt[4]{3} = 3 \sqrt[4]{3} \] **Simplified Form:** \( 3 \sqrt[4]{3} \) --- ### (0) \( \sqrt{3} + \sqrt{27} \) Simplify \( \sqrt{27} \): \[ \sqrt{27} = \sqrt{9 \times 3} = 3 \sqrt{3} \] Now, combine the terms: \[ \sqrt{3} + 3 \sqrt{3} = 4 \sqrt{3} \] **Simplified Form:** \( 4 \sqrt{3} \) --- ### (g) \( 2 \sqrt{18} - \sqrt{32} \) Simplify \( \sqrt{18} \) and \( \sqrt{32} \): \[ \sqrt{18} = \sqrt{9 \times 2} = 3 \sqrt{2} \] \[ \sqrt{32} = \sqrt{16 \times 2} = 4 \sqrt{2} \] Now, substitute back: \[ 2 \times 3 \sqrt{2} - 4 \sqrt{2} = 6 \sqrt{2} - 4 \sqrt{2} = 2 \sqrt{2} \] **Simplified Form:** \( 2 \sqrt{2} \) --- ### (h) \( \frac{\sqrt{32}}{\sqrt{2}} \) Simplify \( \sqrt{32} \): \[ \sqrt{32} = 4 \sqrt{2} \] Now, divide: \[ \frac{4 \sqrt{2}}{\sqrt{2}} = 4 \] **Simplified Form:** \( 4 \) --- ### (i) \( \frac{\sqrt{51}}{\sqrt{3}} \) Simplify the expression: \[ \frac{\sqrt{51}}{\sqrt{3}} = \sqrt{\frac{51}{3}} = \sqrt{17} \] **Simplified Form:** \( \sqrt{17} \) --- ### Summary of Simplified Forms - **(a)** \( \sqrt{21} \) - **(b)** \( 18 \) - **(o)** \( 12 \) - **(d)** \( 9 \sqrt{6} \) - **(c)** \( 3 \sqrt[4]{3} \) - **(0)** \( 4 \sqrt{3} \) - **(g)** \( 2 \sqrt{2} \) - **(h)** \( 4 \) - **(i)** \( \sqrt{17} \)