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Chapter 1

1. $\sqrt{9 + 16} = \sqrt{25} = 5$ $\sqrt{9 + 16} = \sqrt{25} = 5$
2. $\frac{(7) (- 3) (- 2)}{(- 6) (0)}$ $\frac{(7) (- 3) (- 2)}{(- 6) (0)}$ is undefined (division by zero).
3. $\frac{3.372 \times 10^{- 3}}{7.526 \times 10^{12}} = 4.480 \times 10^{- 16}$ $\frac{3.372 \times 10^{- 3}}{7.526 \times 10^{12}} = 4.480 \times 10^{- 16}$
4. $\begin{array}{rcl} \frac{(+ 6) (- 2) - 3 (- 1)}{5 - 2} & = & \frac{- 12 - (- 3)}{3} = \frac{- 12 + 3}{3} \\ = & \frac{- 9}{3} = - 3 \end{array}$ $\begin{array}{rcl} \frac{(+ 6) (- 2) - 3 (- 1)}{5 - 2} & = & \frac{- 12 - (- 3)}{3} = \frac{- 12 + 3}{3} \\ = & \frac{- 9}{3} = - 3 \end{array}$
5. $\frac{346.4 - 23.5}{287.7} - \frac{{0.944}^{3}}{(3.46) (0.109)} = - 1.108$ $\frac{346.4 - 23.5}{287.7} - \frac{{0.944}^{3}}{(3.46) (0.109)} = - 1.108$
6. $\begin{array}{rcl} {(2 a^{0} b^{- 2} c^{3})}^{- 3} & = & 2^{- 3} a^{0 (- 3)} b^{(- 2) (- 3)} c^{3 (- 3)} = 2^{- 3} a^{0} b^{6} c^{- 9} \\ = & \frac{b^{6}}{8 c^{9}} \end{array}$ $\begin{array}{rcl} {(2 a^{0} b^{- 2} c^{3})}^{- 3} & = & 2^{- 3} a^{0 (- 3)} b^{(- 2) (- 3)} c^{3 (- 3)} = 2^{- 3} a^{0} b^{6} c^{- 9} \\ = & \frac{b^{6}}{8 c^{9}} \end{array}$
7. $\begin{array}{rcl} {(2 x + 3)}^{2} & = & (2 x + 3) (2 x + 3) \\ = & 2 x (2 x) + 2 x (3) + 3 (2 x) + 3 (3) \\ = & 4 x^{2} + 6 x + 6 x + 9 \\ = & 4 x^{2} + 12 x + 9 \end{array}$ $\begin{array}{rcl} {(2 x + 3)}^{2} & = & (2 x + 3) (2 x + 3) \\ = & 2 x (2 x) + 2 x (3) + 3 (2 x) + 3 (3) \\ = & 4 x^{2} + 6 x + 6 x + 9 \\ = & 4 x^{2} + 12 x + 9 \end{array}$
8. $\begin{array}{rcl} 3 m^{2} (a m - 2 m^{3}) & = & 3 m^{2} (a m) + 3 m^{2} (- 2 m^{3}) \\ = & 3 a m^{3} - 6 m^{5} \end{array}$ $\begin{array}{rcl} 3 m^{2} (a m - 2 m^{3}) & = & 3 m^{2} (a m) + 3 m^{2} (- 2 m^{3}) \\ = & 3 a m^{3} - 6 m^{5} \end{array}$
9. $\begin{array}{rcl} \frac{8 a^{3} x^{2} - 4 a^{2} x^{4}}{- 2 a x^{2}} & = & \frac{8 a^{3} x^{2}}{- 2 a x^{2}} - \frac{4 a^{2} x^{4}}{- 2 a x^{2}} = - 4 a^{2} - (- 2 a x^{2}) \\ = & - 4 a^{2} + 2 a x^{2} \end{array}$ $\begin{array}{rcl} \frac{8 a^{3} x^{2} - 4 a^{2} x^{4}}{- 2 a x^{2}} & = & \frac{8 a^{3} x^{2}}{- 2 a x^{2}} - \frac{4 a^{2} x^{4}}{- 2 a x^{2}} = - 4 a^{2} - (- 2 a x^{2}) \\ = & - 4 a^{2} + 2 a x^{2} \end{array}$
10. $\begin{array}{l} \begin{array}{l} 3 x - 5 \\ 2 x - 1 6 x^{2} - 13 x + 7 \\ \underline{6 x^{2} - 3 x} \\ - 10 x + 7 \\ \underline{- 10 x + 5} \\ 2 \end{array} & \begin{array}{l} (quotient) \\ (remainder) \end{array} \end{array}$ $\begin{array}{l} \begin{array}{l} 3 x - 5 \\ 2 x - 1 6 x^{2} - 13 x + 7 \\ \underline{6 x^{2} - 3 x} \\ - 10 x + 7 \\ \underline{- 10 x + 5} \\ 2 \end{array} & \begin{array}{l} (quotient) \\ (remainder) \end{array} \end{array}$
11. $\begin{array}{l} (2 x - 3) (x + 7) \\ \begin{array}{cl} = & 2 x (x) + 2 x (7) + (- 3) (x) + (- 3) (7) \\ = & 2 x^{2} + 14 x - 3 x - 21 = 2 x^{2} + 11 x - 21 \end{array} \end{array}$ $\begin{array}{l} (2 x - 3) (x + 7) \\ \begin{array}{cl} = & 2 x (x) + 2 x (7) + (- 3) (x) + (- 3) (7) \\ = & 2 x^{2} + 14 x - 3 x - 21 = 2 x^{2} + 11 x - 21 \end{array} \end{array}$
12. $\begin{array}{rcl} 3 x - [4 x - (3 - 2 x)] & = & 3 x - [4 x - 3 + 2 x] \\ = & 3 x - [6 x - 3] \\ = & 3 x - 6 x + 3 = - 3 x + 3 \end{array}$ $\begin{array}{rcl} 3 x - [4 x - (3 - 2 x)] & = & 3 x - [4 x - 3 + 2 x] \\ = & 3 x - [6 x - 3] \\ = & 3 x - 6 x + 3 = - 3 x + 3 \end{array}$
13. $\begin{array}{rcl} 5 y - 2 (y - 4) & = & 7 \\ 5 y - 2 y + 8 & = & 7 \\ 3 y + 8 - 8 & = & 7 - 8 \\ 3 y & = & - 1 \\ y & = & - 1/3 \end{array}$ $\begin{array}{rcl} 5 y - 2 (y - 4) & = & 7 \\ 5 y - 2 y + 8 & = & 7 \\ 3 y + 8 - 8 & = & 7 - 8 \\ 3 y & = & - 1 \\ y & = & - 1/3 \end{array}$
14. $\begin{array}{rcl} 3 (x - 3) & = & x - (2 - 3 d) \\ 3 x - 9 & = & x - 2 + 3 d \\ 3 x - 9 - x + 9 & = & x - 2 + 3 d - x + 9 \\ 2 x & = & 3 d + 7 \\ x & = & \frac{3 d + 7}{2} \end{array}$ $\begin{array}{rcl} 3 (x - 3) & = & x - (2 - 3 d) \\ 3 x - 9 & = & x - 2 + 3 d \\ 3 x - 9 - x + 9 & = & x - 2 + 3 d - x + 9 \\ 2 x & = & 3 d + 7 \\ x & = & \frac{3 d + 7}{2} \end{array}$
15. $\begin{array}{rcl} 245 lb / {ft}^{3} & = & \frac{245 lb}{1 {ft}^{3}} \times \frac{1 kg}{2.205 lb} \times \frac{1 {ft}^{3}}{28.32 L} \\ = & \frac{245 (1) (1) kg}{1 (2.205) (28.32) L} = 3.92 kg / L \end{array}$ $\begin{array}{rcl} 245 lb / {ft}^{3} & = & \frac{245 lb}{1 {ft}^{3}} \times \frac{1 kg}{2.205 lb} \times \frac{1 {ft}^{3}}{28.32 L} \\ = & \frac{245 (1) (1) kg}{1 (2.205) (28.32) L} = 3.92 kg / L \end{array}$
16. $0.0000036 = 3.6 \times 10^{- 6}$ $0.0000036 = 3.6 \times 10^{- 6}$ (six places to right)

$- π$ $- π$	$- 3$ $- 3$	0.3	$\sqrt{2}$ $\sqrt{2}$	$\| - 4 \|$ $\| - 4 \|$	(order)
$- 3.14$ $- 3.14$	$- 3$ $- 3$	0.3	1.41	4	(value)

18. $3 (5 + 8) = 3 (5) + 3 (8)$ $3 (5 + 8) = 3 (5) + 3 (8)$ illustrates the distributive law.
19.
1. 5,
2. 3.0 (zero is significant)
20. Evaluation: $1000 {(1 + 0.05/2)}^{2 (3)} = 1000 {(1.025)}^{6} = $1159.69$ $1000 {(1 + 0.05/2)}^{2 (3)} = 1000 {(1.025)}^{6} = $1159.69$
21. $\begin{array}{rcl} 8 {(100 - x)}^{2} + x^{2} & = & 8 (100 - x) (100 - x) + x^{2} \\ = & 8 (10, 000 - 200 x + x^{2}) + x^{2} \\ = & 80, 000 - 1600 x + 8 x^{2} + x^{2} \\ = & 80, 000 - 1600 x + 9 x^{2} \end{array}$ $\begin{array}{rcl} 8 {(100 - x)}^{2} + x^{2} & = & 8 (100 - x) (100 - x) + x^{2} \\ = & 8 (10, 000 - 200 x + x^{2}) + x^{2} \\ = & 80, 000 - 1600 x + 8 x^{2} + x^{2} \\ = & 80, 000 - 1600 x + 9 x^{2} \end{array}$
22. $\begin{array}{rcl} L & = & L_{0} [1 + α (t_{2} - t_{1})] \\ L & = & L_{0} [1 + α t_{2} - α t_{1}] \\ L & = & L_{0} + α L_{0} t_{2} - α L_{0} t_{1} \\ L & - & L_{0} + α L_{0} t_{1} = α L_{0} t_{2} \\ t_{2} & = & \frac{L - L_{0} + α L_{0} t_{1}}{α L_{0}} \end{array}$ $\begin{array}{rcl} L & = & L_{0} [1 + α (t_{2} - t_{1})] \\ L & = & L_{0} [1 + α t_{2} - α t_{1}] \\ L & = & L_{0} + α L_{0} t_{2} - α L_{0} t_{1} \\ L & - & L_{0} + α L_{0} t_{1} = α L_{0} t_{2} \\ t_{2} & = & \frac{L - L_{0} + α L_{0} t_{1}}{α L_{0}} \end{array}$
23. Let $n = number$ $n = number$ of pounds of second alloy

$\begin{array}{rcl} 0.3 (20) + 0.8 n & = & 0.6 (n + 20) \\ 6 + 0.8 n & = & 0.6 n + 12 \\ 0.2 n & = & 6 \\ n & = & 30 lb \end{array}$ $\begin{array}{rcl} 0.3 (20) + 0.8 n & = & 0.6 (n + 20) \\ 6 + 0.8 n & = & 0.6 n + 12 \\ 0.2 n & = & 6 \\ n & = & 30 lb \end{array}$

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Table of Contents for Chapter 1

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Table of Contents for
Chapter 1