1)P=(1+x​1​)(x​+11​+x​−11​−x−12​)

\(= \frac{\sqrt{x} + 1}{\sqrt{x}} . \frac{\sqrt{x} - 1 + \sqrt{x} + 1 - 2}{x - 1}\)

\(= \frac{\sqrt{x} + 1}{\sqrt{x}} . \frac{2 \left(\right. \sqrt{x} - 1 \left.\right)}{x - 1}\)

\(= \frac{2}{\sqrt{x}}\).

1)P=(1+x​1​)(x​+11​+x​−11​−x−12​)

\(= \frac{\sqrt{x} + 1}{\sqrt{x}} . \frac{\sqrt{x} - 1 + \sqrt{x} + 1 - 2}{x - 1}\)

\(= \frac{\sqrt{x} + 1}{\sqrt{x}} . \frac{2 \left(\right. \sqrt{x} - 1 \left.\right)}{x - 1}\)

\(= \frac{2}{\sqrt{x}}\).

1)P=(1+x​1​)(x​+11​+x​−11​−x−12​)

\(= \frac{\sqrt{x} + 1}{\sqrt{x}} . \frac{\sqrt{x} - 1 + \sqrt{x} + 1 - 2}{x - 1}\)

\(= \frac{\sqrt{x} + 1}{\sqrt{x}} . \frac{2 \left(\right. \sqrt{x} - 1 \left.\right)}{x - 1}\)

\(= \frac{2}{\sqrt{x}}\).

1)P=(1+x​1​)(x​+11​+x​−11​−x−12​)

\(= \frac{\sqrt{x} + 1}{\sqrt{x}} . \frac{\sqrt{x} - 1 + \sqrt{x} + 1 - 2}{x - 1}\)

\(= \frac{\sqrt{x} + 1}{\sqrt{x}} . \frac{2 \left(\right. \sqrt{x} - 1 \left.\right)}{x - 1}\)

\(= \frac{2}{\sqrt{x}}\).

1)P=(1+x​1​)(x​+11​+x​−11​−x−12​)

\(= \frac{\sqrt{x} + 1}{\sqrt{x}} . \frac{\sqrt{x} - 1 + \sqrt{x} + 1 - 2}{x - 1}\)

\(= \frac{\sqrt{x} + 1}{\sqrt{x}} . \frac{2 \left(\right. \sqrt{x} - 1 \left.\right)}{x - 1}\)

\(= \frac{2}{\sqrt{x}}\).

1)P=(1+x​1​)(x​+11​+x​−11​−x−12​)

\(= \frac{\sqrt{x} + 1}{\sqrt{x}} . \frac{\sqrt{x} - 1 + \sqrt{x} + 1 - 2}{x - 1}\)

\(= \frac{\sqrt{x} + 1}{\sqrt{x}} . \frac{2 \left(\right. \sqrt{x} - 1 \left.\right)}{x - 1}\)

\(= \frac{2}{\sqrt{x}}\).

1)P=(1+x​1​)(x​+11​+x​−11​−x−12​)

\(= \frac{\sqrt{x} + 1}{\sqrt{x}} . \frac{\sqrt{x} - 1 + \sqrt{x} + 1 - 2}{x - 1}\)

\(= \frac{\sqrt{x} + 1}{\sqrt{x}} . \frac{2 \left(\right. \sqrt{x} - 1 \left.\right)}{x - 1}\)

\(= \frac{2}{\sqrt{x}}\).

) \(2 \sqrt{\frac{2}{3}} - 4 \sqrt{\frac{3}{2}}\)

\(= 2 \sqrt{\frac{2.3}{3^{2}}} - 4 \sqrt{\frac{3.2}{2^{2}}}\)

\(= 2. \frac{\sqrt{6}}{3} - 2. \sqrt{6}\)

\(= - \frac{4 \sqrt{6}}{3}\).

b) \(\frac{5 \sqrt{48} - 3 \sqrt{27} + 2 \sqrt{12}}{\sqrt{3}}\)

\(= \frac{5 \sqrt{4^{2} . 3} - 3 \sqrt{3^{2} . 3} + 2 \sqrt{2^{2} . 3}}{\sqrt{3}}\)

\(= \frac{20 \sqrt{3} - 9 \sqrt{3} + 4 \sqrt{3}}{\sqrt{3}}\)

\(= \frac{15 \sqrt{3}}{\sqrt{3}} = 15\)

c) \(\frac{1}{3 + 2 \sqrt{2}} + \frac{4 \sqrt{2} - 4}{2 - \sqrt{2}}\)

\(= \frac{1. \left(\right. 3 - 2 \sqrt{2} \left.\right)}{\left(\right. 3 + 2 \sqrt{2} \left.\right) \left(\right. 3 - 2 \sqrt{2} \left.\right)} + \frac{4 \left(\right. \sqrt{2} - 1 \left.\right)}{\sqrt{2} \left(\right. \sqrt{2} - 1 \left.\right)}\)

\(= \frac{3 - 2 \sqrt{2}}{9 - 8} + \frac{4}{\sqrt{2}}\)

\(= 3 - 2 \sqrt{2} + 2 \sqrt{2}\)

\(= 3\)

) \(2 \sqrt{\frac{2}{3}} - 4 \sqrt{\frac{3}{2}}\)

\(= 2 \sqrt{\frac{2.3}{3^{2}}} - 4 \sqrt{\frac{3.2}{2^{2}}}\)

\(= 2. \frac{\sqrt{6}}{3} - 2. \sqrt{6}\)

\(= - \frac{4 \sqrt{6}}{3}\).

b) \(\frac{5 \sqrt{48} - 3 \sqrt{27} + 2 \sqrt{12}}{\sqrt{3}}\)

\(= \frac{5 \sqrt{4^{2} . 3} - 3 \sqrt{3^{2} . 3} + 2 \sqrt{2^{2} . 3}}{\sqrt{3}}\)

\(= \frac{20 \sqrt{3} - 9 \sqrt{3} + 4 \sqrt{3}}{\sqrt{3}}\)

\(= \frac{15 \sqrt{3}}{\sqrt{3}} = 15\)

c) \(\frac{1}{3 + 2 \sqrt{2}} + \frac{4 \sqrt{2} - 4}{2 - \sqrt{2}}\)

\(= \frac{1. \left(\right. 3 - 2 \sqrt{2} \left.\right)}{\left(\right. 3 + 2 \sqrt{2} \left.\right) \left(\right. 3 - 2 \sqrt{2} \left.\right)} + \frac{4 \left(\right. \sqrt{2} - 1 \left.\right)}{\sqrt{2} \left(\right. \sqrt{2} - 1 \left.\right)}\)

\(= \frac{3 - 2 \sqrt{2}}{9 - 8} + \frac{4}{\sqrt{2}}\)

\(= 3 - 2 \sqrt{2} + 2 \sqrt{2}\)

\(= 3\)

) \(2 \sqrt{\frac{2}{3}} - 4 \sqrt{\frac{3}{2}}\)

\(= 2 \sqrt{\frac{2.3}{3^{2}}} - 4 \sqrt{\frac{3.2}{2^{2}}}\)

\(= 2. \frac{\sqrt{6}}{3} - 2. \sqrt{6}\)

\(= - \frac{4 \sqrt{6}}{3}\).

b) \(\frac{5 \sqrt{48} - 3 \sqrt{27} + 2 \sqrt{12}}{\sqrt{3}}\)

\(= \frac{5 \sqrt{4^{2} . 3} - 3 \sqrt{3^{2} . 3} + 2 \sqrt{2^{2} . 3}}{\sqrt{3}}\)

\(= \frac{20 \sqrt{3} - 9 \sqrt{3} + 4 \sqrt{3}}{\sqrt{3}}\)

\(= \frac{15 \sqrt{3}}{\sqrt{3}} = 15\)

c) \(\frac{1}{3 + 2 \sqrt{2}} + \frac{4 \sqrt{2} - 4}{2 - \sqrt{2}}\)

\(= \frac{1. \left(\right. 3 - 2 \sqrt{2} \left.\right)}{\left(\right. 3 + 2 \sqrt{2} \left.\right) \left(\right. 3 - 2 \sqrt{2} \left.\right)} + \frac{4 \left(\right. \sqrt{2} - 1 \left.\right)}{\sqrt{2} \left(\right. \sqrt{2} - 1 \left.\right)}\)

\(= \frac{3 - 2 \sqrt{2}}{9 - 8} + \frac{4}{\sqrt{2}}\)

\(= 3 - 2 \sqrt{2} + 2 \sqrt{2}\)

\(= 3\)