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1.a 3^6 x 3^2 : 3^5
= 3^6+2-5
= 3^3 = 27

b. 2^6 : (2^3)^6
= 2^6 : 2^3x6
= 2^6 : 2^18
= 2^6-18
=2^-12
= 1/2^12

c. (6^2 :6^20):6^1
= 6^2-20 :6^1
= 6^-18 : 6^1
= 6^-18-1
= 6^-19

[tex]2a. \: \sqrt{300 } \div \sqrt{6 } \\ = \sqrt{ \frac{300}{6} } \\ = \sqrt{50} \\ = \sqrt{25 \times 2} \\ = 5 \sqrt{2} [/tex]
[tex]2b. \: \sqrt{48} + \sqrt{125} \\ = \sqrt{16 \times 3} + \sqrt{25 \times 5} \\ = 4 \sqrt{3} + 5 \sqrt{5} [/tex]
[tex]2c \: \sqrt{20} - \sqrt{45} + \sqrt{75} \\ = \sqrt{4 \times 5} - \sqrt{9 \times 5} + \sqrt{25 \times 3} \\ = 2 \sqrt{5} - 3 \sqrt{5} + 5 \sqrt{3} \\ = (2 - 3) \sqrt{5} + 5 \sqrt{3} \\ = - 1 \sqrt{5} + 5 \sqrt{3} [/tex]
[tex]3a \: \sqrt{7 {}^{5} } = {7}^{ \frac{5}{2} } \\ \\ 3b \: \sqrt[3]{ {2}^{2} } = {2}^{ \frac{2}{3} } \\ \\ 3c \: \sqrt{ {6}^{3} } = {6}^{ \frac{3}{2} } [/tex]
[tex]4a \: \frac{2}{ \sqrt{6} } = \frac{2}{ \sqrt{6} } \times \frac{ \sqrt{6} }{ \sqrt{6} } \\ = \frac{2}{6} \sqrt{6} \\ = \frac{1}{3} \sqrt{6} \\[/tex]
[tex]4b \: \frac{3 \sqrt{2} }{ \sqrt{6} } = \frac{3 \sqrt{2} }{ \sqrt{6} } \times \frac{ \sqrt{6} }{ \sqrt{6} } \\ = \frac{3 \sqrt{2 \times 6} }{6} \\ = \frac{1}{3} \sqrt{12} \\ = \frac{1}{3} \sqrt{4 \times 3} \\ = \frac{2}{3} \sqrt{3} [/tex]
[tex]4c \: \frac{3 + \sqrt{2} }{3 - \sqrt{2} } = \frac{(3 + \sqrt{2}) (3 + \sqrt{2} }{ {3}^{2} - 2 } \\ = \frac{9 + 6 \sqrt{2} + 2}{7} \\ = \frac{11 + 6 \sqrt{2} }{7} [/tex]