Chapter 11

1. $- 5 y^{0} = - 5 (1) = - 5$ $- 5 y^{0} = - 5 (1) = - 5$
2. ${(3 π x^{- 4})}^{- 2} = {(3 π)}^{- 2} {(x^{- 4})}^{- 2} = \frac{x^{8}}{{(3 π)}^{2}} = \frac{x^{8}}{9 π^{2}}$ ${(3 π x^{- 4})}^{- 2} = {(3 π)}^{- 2} {(x^{- 4})}^{- 2} = \frac{x^{8}}{{(3 π)}^{2}} = \frac{x^{8}}{9 π^{2}}$
3. $\begin{array}{rcl} \frac{100^{3/2}}{8^{- 2/3}} & = & (100^{3/2}) (8^{2/3}) = [{(100^{1/2})}^{3}] [{(8^{1/3})}^{2}] \\ = & (10^{3}) (2^{2}) = 4000 \end{array}$ $\begin{array}{rcl} \frac{100^{3/2}}{8^{- 2/3}} & = & (100^{3/2}) (8^{2/3}) = [{(100^{1/2})}^{3}] [{(8^{1/3})}^{2}] \\ = & (10^{3}) (2^{2}) = 4000 \end{array}$
4. ${(a s^{- 1/3} t^{3/4})}^{12} = a^{12} s^{(- 1/3) (12)} t^{(3/4) (12)} = a^{12} s^{- 4} t^{9} = \frac{a^{12} t^{9}}{s^{4}}$ ${(a s^{- 1/3} t^{3/4})}^{12} = a^{12} s^{(- 1/3) (12)} t^{(3/4) (12)} = a^{12} s^{- 4} t^{9} = \frac{a^{12} t^{9}}{s^{4}}$
5. $\begin{array}{rcl} 2 \sqrt{20} - \sqrt{125} & = & 2 \sqrt{4 \times 5} - \sqrt{25 \times 5} \\ = & 2 (2 \sqrt{5}) - 5 \sqrt{5} \\ = & 4 \sqrt{5} - 5 \sqrt{5} = - \sqrt{5} \end{array}$ $\begin{array}{rcl} 2 \sqrt{20} - \sqrt{125} & = & 2 \sqrt{4 \times 5} - \sqrt{25 \times 5} \\ = & 2 (2 \sqrt{5}) - 5 \sqrt{5} \\ = & 4 \sqrt{5} - 5 \sqrt{5} = - \sqrt{5} \end{array}$
6. $\begin{array}{rcl} {(2 x^{- 1} + y^{- 2})}^{- 1} & = & \frac{1}{2 x^{- 1} + y^{- 2}} = \frac{1}{\frac{2}{x} + \frac{1}{y^{2}}} = \frac{1}{\frac{2 y^{2} + x}{x y^{2}}} \\ = & \frac{x y^{2}}{2 y^{2} + x} \end{array}$ $\begin{array}{rcl} {(2 x^{- 1} + y^{- 2})}^{- 1} & = & \frac{1}{2 x^{- 1} + y^{- 2}} = \frac{1}{\frac{2}{x} + \frac{1}{y^{2}}} = \frac{1}{\frac{2 y^{2} + x}{x y^{2}}} \\ = & \frac{x y^{2}}{2 y^{2} + x} \end{array}$
7. $\begin{array}{rcl} {(\sqrt{2 x} - 3 \sqrt{y})}^{2} & = & {(\sqrt{2 x})}^{2} - 2 \sqrt{2 x} (3 \sqrt{y}) + {(3 \sqrt{y})}^{2} \\ = & 2 x - 6 \sqrt{2 x y} + 9 y \end{array}$ $\begin{array}{rcl} {(\sqrt{2 x} - 3 \sqrt{y})}^{2} & = & {(\sqrt{2 x})}^{2} - 2 \sqrt{2 x} (3 \sqrt{y}) + {(3 \sqrt{y})}^{2} \\ = & 2 x - 6 \sqrt{2 x y} + 9 y \end{array}$
8. $\sqrt[3]{\sqrt[4]{4}} = \sqrt[12]{4} = \sqrt[12]{2^{2}} = 2^{2/12} = 2^{1/6} = \sqrt[6]{2}$ $\sqrt[3]{\sqrt[4]{4}} = \sqrt[12]{4} = \sqrt[12]{2^{2}} = 2^{2/12} = 2^{1/6} = \sqrt[6]{2}$
9. $\frac{3 - 2 \sqrt{2}}{2 \sqrt{x}} = \frac{3 - 2 \sqrt{2}}{2 \sqrt{x}} \times \frac{\sqrt{x}}{\sqrt{x}} = \frac{3 \sqrt{x} - 2 \sqrt{2 x}}{2 x}$ $\frac{3 - 2 \sqrt{2}}{2 \sqrt{x}} = \frac{3 - 2 \sqrt{2}}{2 \sqrt{x}} \times \frac{\sqrt{x}}{\sqrt{x}} = \frac{3 \sqrt{x} - 2 \sqrt{2 x}}{2 x}$
10. $\sqrt{27 a^{4} b^{3}} = \sqrt{9 \times 3 \times {(a^{2})}^{2} (b^{2}) (b)} = 3 a^{2} b \sqrt{3 b}$ $\sqrt{27 a^{4} b^{3}} = \sqrt{9 \times 3 \times {(a^{2})}^{2} (b^{2}) (b)} = 3 a^{2} b \sqrt{3 b}$
11. $\begin{array}{rcl} 2 \sqrt{2} (3 \sqrt{10} - \sqrt{6}) & = & 2 \sqrt{2} (3 \sqrt{10}) - 2 \sqrt{2} (\sqrt{6}) \\ = & 6 \sqrt{20} - 2 \sqrt{12} \\ = & 6 (2 \sqrt{5}) - 2 (2 \sqrt{3}) \\ = & 12 \sqrt{5} - 4 \sqrt{3} \end{array}$ $\begin{array}{rcl} 2 \sqrt{2} (3 \sqrt{10} - \sqrt{6}) & = & 2 \sqrt{2} (3 \sqrt{10}) - 2 \sqrt{2} (\sqrt{6}) \\ = & 6 \sqrt{20} - 2 \sqrt{12} \\ = & 6 (2 \sqrt{5}) - 2 (2 \sqrt{3}) \\ = & 12 \sqrt{5} - 4 \sqrt{3} \end{array}$
12. $\begin{array}{l} {(2 x + 3)}^{1/2} + (x + 1) {(2 x + 3)}^{- 1/2} \\ \begin{array}{cl} = & {(2 x + 3)}^{1/2} + \frac{x + 1}{{(2 x + 3)}^{1/2}} \\ = & \frac{{(2 x + 3)}^{1/2} {(2 x + 3)}^{1/2} + x + 1}{{(2 x + 3)}^{1/2}} = \frac{2 x + 3 + x + 1}{{(2 x + 3)}^{1/2}} \\ = & \frac{3 x + 4}{{(2 x + 3)}^{1/2}} \end{array} \end{array}$ $\begin{array}{l} {(2 x + 3)}^{1/2} + (x + 1) {(2 x + 3)}^{- 1/2} \\ \begin{array}{cl} = & {(2 x + 3)}^{1/2} + \frac{x + 1}{{(2 x + 3)}^{1/2}} \\ = & \frac{{(2 x + 3)}^{1/2} {(2 x + 3)}^{1/2} + x + 1}{{(2 x + 3)}^{1/2}} = \frac{2 x + 3 + x + 1}{{(2 x + 3)}^{1/2}} \\ = & \frac{3 x + 4}{{(2 x + 3)}^{1/2}} \end{array} \end{array}$
13. $(\frac{4 a^{- 1/2} b^{3/4}}{b^{- 2}}) (\frac{b^{- 1}}{2 a}) = \frac{4 a^{- 1/2} b^{3/4 - 1}}{2 a b^{- 2}} = \frac{2 b^{2 - 1/4}}{a^{1 + 1/2}} = \frac{2 b^{7/4}}{a^{3/2}}$ $(\frac{4 a^{- 1/2} b^{3/4}}{b^{- 2}}) (\frac{b^{- 1}}{2 a}) = \frac{4 a^{- 1/2} b^{3/4 - 1}}{2 a b^{- 2}} = \frac{2 b^{2 - 1/4}}{a^{1 + 1/2}} = \frac{2 b^{7/4}}{a^{3/2}}$
14. $\begin{array}{rcl} \frac{2 \sqrt{15} + \sqrt{3}}{\sqrt{15} - 2 \sqrt{3}} & = & \frac{2 \sqrt{15} + \sqrt{3}}{\sqrt{15} - 2 \sqrt{3}} \times \frac{\sqrt{15} + 2 \sqrt{3}}{\sqrt{15} + 2 \sqrt{3}} \\ = & \frac{2 (15) + 4 \sqrt{45} + \sqrt{45} + 2 (3)}{15 - 4 (3)} \\ = & \frac{36 + 15 \sqrt{5}}{3} \\ = & 12 + 5 \sqrt{5} \end{array}$ $\begin{array}{rcl} \frac{2 \sqrt{15} + \sqrt{3}}{\sqrt{15} - 2 \sqrt{3}} & = & \frac{2 \sqrt{15} + \sqrt{3}}{\sqrt{15} - 2 \sqrt{3}} \times \frac{\sqrt{15} + 2 \sqrt{3}}{\sqrt{15} + 2 \sqrt{3}} \\ = & \frac{2 (15) + 4 \sqrt{45} + \sqrt{45} + 2 (3)}{15 - 4 (3)} \\ = & \frac{36 + 15 \sqrt{5}}{3} \\ = & 12 + 5 \sqrt{5} \end{array}$
15. $\begin{array}{rcl} \frac{3^{- 1/2}}{2} & = & \frac{1}{2 \times 3^{1/2}} = \frac{1}{2 \sqrt{3}} \\ = & \frac{\sqrt{3}}{2 \sqrt{3} \sqrt{3}} = \frac{\sqrt{3}}{6} \end{array}$ $\begin{array}{rcl} \frac{3^{- 1/2}}{2} & = & \frac{1}{2 \times 3^{1/2}} = \frac{1}{2 \sqrt{3}} \\ = & \frac{\sqrt{3}}{2 \sqrt{3} \sqrt{3}} = \frac{\sqrt{3}}{6} \end{array}$
16. $\begin{array}{l} 0.220 N^{- 1/6} N = 64 \times 10^{6} \\ 0.220 {(64 \times 10^{6})}^{- 1/6} = \frac{0.220}{{(64 \times 10^{6})}^{1/6}} = \frac{0.220}{2 \times 10} = 0.011 \end{array}$ $\begin{array}{l} 0.220 N^{- 1/6} N = 64 \times 10^{6} \\ 0.220 {(64 \times 10^{6})}^{- 1/6} = \frac{0.220}{{(64 \times 10^{6})}^{1/6}} = \frac{0.220}{2 \times 10} = 0.011 \end{array}$

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Table of Contents for Chapter 11

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Table of Contents for
Chapter 11