(i)

$$ 2020, 6 $$

(ii)

$$ -5, -17 $$

(iii)

$$ 2020, 1.666, \frac{3}{4}, 6 $$

(iv)

$$ -5, - \frac{1}{2}, -3.8, -17, - \frac{5}{3} $$

(a)

$$ -43.6^\circ \text{C} $$

(b)

$$ -10 \, 994 \text{ m} $$

(c)

$$ - \$ 10 \, 000 $$

(d)

$$ -81^\circ $$

(b)

$$ -7 < - 2 $$

(c)

$$ 4 > -4 $$

(d)

$$ -6 > -100 $$

(a)

$$ (-4) + (-5) = -9 $$

(b)

$$ -9 + (-7) = -16 $$

(c)

$$ (-21) + (-73) = -94 $$

(d)

$$ (-67) + (-48) = -115 $$

(a)

$$ 8 + (-3) = 8 - 3 = 5 $$

(b)

$$ 45 + (-17) = 45 - 17 = 28 $$

(c)

$$ -2 + 6 = 6 - 2 = 4 $$

(d)

$$ -12 + 35 = 35 - 12 = 23 $$

(a)

$$ 5 - 9 = -4 $$

(b)

$$ 38 - 59 = -21 $$

(c)

$$ 8 + (-11) = 8 - 11 = -3 $$

(d)

$$ (-92) + 47 = 47 - 92 = -45 $$

(a)

$$ -9 - 11 = -20 $$

(b)

$$ -39 - 12 = -51 $$

(c)

$$ -146 - 218 = -364 $$

(a)

$$ 4 - (-6) = 4 + 6 = 10 $$

(b)

$$ 30 - (-16) = 30 + 16 = 46 $$

(c)

$$ -8 - (-3) = -8 + 3 = -5 $$

(d)

$$ -17 - (-35) = -17 + 35 = 18 $$

\begin{align*} \text{Temperature at } A & = -5^\circ \text{C} \\ \text{Temperature at } B & = 23^\circ \text{C} \\ \\ \text{Difference in temperature} & = [23 - (-5)]^\circ \text{C} \\ & = (23 + 5)^\circ \text{C} \\ & = 28^\circ \text{C} \end{align*}

\begin{align*} \text{Altitude at } D & = -165 \text{ m} \\ \\ \text{Difference in altitude} & = [314 - (-165)] \text{ m} \\ & = (314 + 165) \text{ m} \\ & = 479 \text{ m} \end{align*}

(a)

$$ 2 \times (-9) = -18 $$

(b)

$$ (-8) \times 4 = -32 $$

(c)

$$ (-6)(-7) = (-6) \times (-7) = 42 $$

(d)

$$ -(-10) = 10 $$

(e)

$$ 11 \times (-3) = -33 $$

(f)

$$ -19 \times 10 = -190 $$

(g)

$$ -12 \times (-3) = 36 $$

(h)

$$ -4(-7) = -4 \times (-7) = 28 $$

(i)

$$ 2020 \times (-1) = -2020 $$

(a)

$$ (-8) \div 2 = -4 $$

(b)

$$ 15 \div (-3) = -5 $$

(c)

$$ -21 \div (-7) = 3 $$

(d)

$$ \frac{-16}{4} = -4 $$

(e)

$$ \frac{20}{-5} = -4 $$

(f)

$$ \frac{-24}{-3} = 8 $$

(a)

$$ \pm 1, \pm 2, \pm 4, \pm 8 $$

(b)

$$ \pm 1, \pm 3, \pm 9 $$

(c)

$$ \pm 1, \pm 7 $$

(d)

$$ \pm 1 $$

One possible pair of multiples for each part is:

(a)

$$ 4, -4 $$

(b)

$$ 12, -12 $$

(c)

$$ 5, -5 $$

(d)

$$ 6, -6 $$

(a)

$$ \text{Square roots of } 64 = \pm \sqrt{64} = \pm \sqrt{8 \times 8} = \pm 8 $$

(b)

$$ \text{Negative square root of } 9 = -\sqrt{9} = - \sqrt{3 \times 3} = -3 $$

(c)

$$ \sqrt{36} = 6 $$

(a)

$$ (-3)^3 = (-3) \times (-3) \times (-3) = -27 $$

(b)

$$ \sqrt[3]{64} = \sqrt[3]{4^3} = \sqrt[3]{4 \times 4 \times 4} = 4 $$

(c)

$$ \sqrt[3]{-8} = \sqrt[3]{(-2)^3} = \sqrt[3]{(-2) \times (-2) \times (-2)} = -2 $$

(d)

$$ -\sqrt[3]{-27} = -\sqrt[3]{(-3) \times (-3) \times (-3)} = 3 $$

(a)

\begin{align*} -2 \times (15 - \sqrt{49} + 2^3) & = -2 \times (15 - \sqrt{7 \times 7} + 2 \times 2 \times 2) \\ & = -2 \times (15 - 7 + 8) \\ & = -2 \times (8 + 8) \\ & = -2 \times 16 \\ & = -32 \end{align*}

(b)

\begin{align*} 4^3 - \{7 \times [16 - (\sqrt[3]{64} - 5)]\} & = 4 \times 4 \times 4 - \{7 \times [16 - (\sqrt[3]{4 \times 4 \times 4} - 5)]\} \\ & = 64 - \{7 \times [16 - (4 - 5)]\} \\ & = 64 - \{7 \times [16 - (-1)]\} \\ & = 64 - \{7 \times (16 + 1)\} \\ & = 64 - (7 \times 17) \\ & = 64 - 119 \\ & = -55 \end{align*}

(a)

\begin{align*} 7 \frac{1}{2} + \left(-3 \frac{3}{5}\right) & = \frac{15}{2} - \frac{18}{5} \\ & = \frac{75}{10} - \frac{36}{10} \\ & = \frac{39}{10} \\ & = 3 \frac{9}{10} \end{align*}

(b)

\begin{align*} -3 \frac{1}{3} + \left(-\frac{1}{4}\right) & = -\frac{10}{3} - \frac{1}{4} \\ & = -\frac{40}{12} - \frac{3}{12} \\ & = -\frac{43}{12} \\ & = -3 \frac{7}{12} \end{align*}

(c)

\begin{align*} -2 \frac{3}{4} - \frac{5}{6} - \left(-\frac{8}{3}\right) & = -\frac{11}{4} - \frac{5}{6} + \frac{8}{3} \\ & = -\frac{33}{12} - \frac{10}{12} + \frac{32}{12} \\ & = \frac{-33 - 10 + 32}{12} \\ & = -\frac{11}{12} \end{align*}

(d)

\begin{align*} \frac{5}{9} - 2 \frac{1}{6} - \left(-\frac{4}{3}\right) & = \frac{5}{9} - \frac{13}{6} + \frac{4}{3} \\ & = \frac{10}{18} - \frac{39}{18} + \frac{24}{18} \\ & = \frac{10 - 39 + 24}{18} \\ & = -\frac{5}{18} \end{align*}

(a)

\begin{align*} 2 \frac{2}{7} \times \frac{7}{16} & = \frac{16}{7} \times \frac{7}{16} \\ & = 1 \end{align*}

(b)

\begin{align*} \frac{20}{3} \times \frac{21}{5} & = 4 \times 7 \\ & = 28 \end{align*}

(c)

\begin{align*} \frac{11}{19} \times 1 \frac{9}{22} & = \frac{11}{19} \times \frac{31}{22} \\ & = \frac{1}{19} \times \frac{31}{2} \\ & = \frac{31}{38} \end{align*}

(d)

\begin{align*} 8 \frac{1}{8} \times 1 \frac{5}{13} & = \frac{65}{8} \times \frac{18}{13} \\ & = \frac{5}{4} \times 9 \\ & = \frac{45}{4} \\ & = 11 \frac{1}{4} \end{align*}

(a)

\begin{align*} \frac{12}{5} \div \frac{7}{3} & = \frac{12}{5} \times \frac{3}{7} \\ & = \frac{36}{35} \\ & = 1 \frac{1}{35} \end{align*}

(b)

\begin{align*} 2 \frac{7}{10} \div \frac{6}{25} & = \frac{27}{10} \div \frac{6}{25} \\ & = \frac{27}{10} \times \frac{25}{6} \\ & = \frac{9}{2} \times \frac{5}{2} \\ & = \frac{45}{4} \\ & = 11 \frac{1}{4} \end{align*}

(c)

\begin{align*} 6 \frac{3}{7} \div 5 & = \frac{45}{7} \div 5 \\ & = \frac{45}{7} \times \frac{1}{5} \\ & = \frac{9}{7} \\ & = 1 \frac{2}{7} \end{align*}

(d)

\begin{align*} 4 \frac{4}{5} \div 1 \frac{1}{15} & = \frac{24}{5} \div \frac{16}{15} \\ & = \frac{24}{5} \times \frac{15}{16} \\ & = \frac{3}{2} \times 3 \\ & = \frac{9}{2} \\ & = 4 \frac{1}{2} \end{align*}

(a)

\begin{align*} 5 \frac{1}{4} \div \left(-2 \frac{4}{5}\right) & = \frac{21}{4} \div \left(-\frac{14}{5}\right) \\ & = \frac{21}{4} \times \left(-\frac{5}{14}\right) \\ & = \frac{3}{4} \times \left(-\frac{5}{2}\right) \\ & = -\frac{15}{8} \\ & = -1 \frac{7}{8} \end{align*}

(b)

\begin{align*} -\frac{4}{7} \times \left[\left(\frac{1}{2}\right)^2 - \left(-1 \frac{3}{4}\right)\right] & = -\frac{4}{7} \times \left[\frac{1}{2} \times \frac{1}{2} + \frac{7}{4}\right] \\ & = -\frac{4}{7} \times \left(\frac{1}{4} + \frac{7}{4}\right) \\ & = -\frac{4}{7} \times 2 \\ & = -\frac{8}{7} \\ & = -1 \frac{1}{7} \end{align*}

(c)

\begin{align*} \left[-\frac{3}{4} + \left(-\frac{5}{2}\right)\right] \div \frac{2}{13} & = \left(-\frac{3}{4} - \frac{10}{4}\right) \div \frac{2}{13} \\ & = -\frac{13}{4} \div \frac{2}{13} \\ & = -\frac{13}{4} \times \frac{13}{2} \\ & = -\frac{169}{8} \\ & = -21 \frac{1}{8} \end{align*}

(d)

\begin{align*} -\frac{9}{14} \times \left[\left(-\frac{3}{2}\right)^2 - 2 \frac{5}{6}\right] & = -\frac{9}{14} \times \left[\left(-\frac{3}{2}\right) \times \left(-\frac{3}{2}\right) - \frac{17}{6}\right] \\ & = -\frac{9}{14} \times \left(\frac{9}{4} - \frac{17}{6}\right) \\ & = -\frac{9}{14} \times \left(\frac{27}{12} - \frac{34}{12}\right) \\ & = -\frac{9}{14} \times \left(-\frac{7}{12}\right) \\ & = \frac{3}{8} \end{align*}

(a)

\begin{align*} -93.8 - 7.236 & = -(93.800 + 7.236) \\ & = -101.036 \end{align*}

(b)

\begin{align*} 5.4 - 15 & = -(15.0 - 5.4) \\ & = -9.6 \end{align*}

(c)

\begin{align*} 124.8 + (-7.24) - (-22.44) & = 124.80 - 7.24 + 22.44 \\ & = 117.56 + 22.44 \\ & = 140 \end{align*}

(a)

\begin{align*} 13.56 \times 2.4 & = \frac{1356}{100} \times \frac{24}{10} \\ & = \frac{1356 \times 24}{1000} \\ & = \frac{32\,544}{1000} \\ & = 32.544 \end{align*}

(b)

\begin{align*} 137.8 \times 0.351 & = \frac{1378}{10} \times \frac{351}{1000} \\ & = \frac{1378 \times 351}{10\,000} \\ & = \frac{483\,678}{10\,000} \\ & = 48.3678 \end{align*}

(a)

\begin{align*} 0.92 \div 0.4 & = \frac{0.92}{0.4} \\ & = \frac{0.92 \times 10}{0.4 \times 10} \\ & = \frac{9.2}{4} \\ & = 2.3 \end{align*}

(b)

\begin{align*} 1.845 \div 0.15 & = \frac{1.845}{0.15} \\ & = \frac{1.845 \times 100}{0.15 \times 100} \\ & = \frac{184.5}{15} \\ & = 12.3 \end{align*}

(a)

\begin{align*} \frac{0.12}{0.4} \times \left(\frac{-0.23}{0.6}\right) & = \frac{1.2}{4} \times \left(\frac{-2.3}{6}\right) \\ & = 0.3 \times \left(\frac{-2.3}{6}\right) \\ & = \frac{0.3 \times (-2.3)}{6} \\ & = \frac{-0.69}{6} \\ & = -0.115 \end{align*}

(b)

\begin{align*} -7.2 \times [-1.3 + (-3.5)] & = -7.2 \times (-4.8) \\ & = 7.2 \times 4.8 \\ & = 34.56 \end{align*}

(c)

\begin{align*} -0.3^2 \times 4.5 \div (-2.7) - 0.65 & = -(0.3 \times 0.3) \times 4.5 \div (-2.7) - 0.65 \\ & = -0.09 \times 4.5 \div (-2.7) - 0.65 \\ & = -0.405 \div (-2.7) - 0.65 \\ & = 0.15 - 0.65 \\ & = -0.5 \end{align*}

(d)

\begin{align*} 3.2^3 \times (-0.625)^2 \div (-6.4) & = (3.2 \times 3.2 \times 3.2) \times [(-0.625) \times (-0.625)] \div (-6.4) \\ & = 32.768 \times 0.390625 \div (-6.4) \\ & = 12.8 \div (-6.4) \\ & = -2 \end{align*}

\begin{align*} \frac{\pi \times 0.7^2}{\sqrt[3]{2.4} + 1 \frac{3}{10}} & = \frac{0.49\pi}{\sqrt[3]{2.4} + 1.3} \\ & \approx 0.5833492335 \\ & \approx 0.583 \text{ (3 d.p.)} \end{align*}

(i)

$$ 10 \, 001, 4 $$

(ii)

$$ -12, -2026 $$

(iii)

$$ \frac{1}{5}, 4.33, 10 \, 001, 4 $$

(iv)

$$ -0.3, - \frac{9}{7}, -12, -1 \frac{1}{2}, -2026 $$

(a)

$$ -273.15^\circ \text{C} $$

(b)

$$ -86 \text{ m} $$

(c)

$$ - \$ 85 \, 000 $$

(d)

$$ -0.15 \text{ kg} $$

(a)

$$ 16 < 60 $$

(b)

$$ 3.1 < 3.2 $$

(c)

$$ -6 < 8 $$

(d)

$$ 30 > -31 $$

(e)

$$ -2 < 0 $$

(f)

$$ 9.8 > -9.9 $$

(a)

$$ -4, -2.8, 0, 2 \frac{2}{5}, 6 $$

(b)

$$ -2, -0.55, - \frac{1}{10}, 2, 4 $$

(c)

$$ -3, -2, -1, 0, 1, 2, 3 $$

(d)

$$ 1, 2, 3, 4, 5, 6, 7, 8, 9 $$

(a)

$$ -13, -3, 23, 30, 230 $$

(b)

$$ -10, -0.5, - \frac{3}{20}, 15, 150 $$

(a)

An altitude of \(10\) m below sea level.

(b)

A withdrawal of \(\$25\) from the bank account.

(c)

An anticlockwise rotation of \(30^\circ\).

(d)

A deduction of \(4\) marks in a test.

(a)

$$ -4 > -6 $$

(b)

$$ -11 > -11.5 $$

(c)

$$ \frac{1}{5} < \frac{1}{3} $$

(d)

$$ - \frac{1}{3} > - \frac{5}{6} $$

(a)

$$ - \frac{1}{3}, -0.2, 0.11, 1, 1 \frac{3}{8}, 2.5 $$

(b)

$$ 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 $$

(c)

$$ 2, 3, 5, 7 $$

(d)

\begin{align*} \text{Factors of } 12 & = 1, 2, 3, 4, 6, 12 \\ \\ \text{Factors of } 16 & = 1, 2, 4, 8, 16 \\ \\ \text{Common factors of 12 and 16} & = 1, 2, 4 \end{align*}

(a)

$$ -106, -107 $$

(b)

$$ 0, 6, 7 $$

(c)

$$ -7, -6, -5, -4 $$

(d)

$$ -4.76, -4.67 $$

(a)

$$ (-8) + (-3) = -11 $$

(b)

$$ -1 + (-6) = -7 $$

(c)

$$ (-4) + (-7) = -11 $$

(d)

$$ -5 + (-2) = -7 $$

(a)

$$ 9 + (-4) = 9 - 4 = 5 $$

(b)

$$ 10 + (-3) = 10 - 3 = 7 $$

(c)

$$ -17 + 29 = 29 - 17 = 12 $$

(d)

$$ -20 + 20 = 0 $$

(a)

$$ 3 - 8 = -5 $$

(b)

$$ 0 - 11 = -11 $$

(c)

$$ 7 - 14 = -7 $$

(d)

$$ 5 + (-8) = 5 - 8 = -3 $$

(e)

$$ 0 + (-19) = -19 $$

(f)

$$ -12 + 7 = 7 - 12 = -5 $$

(a)

$$ -2 - 7 = -9 $$

(b)

$$ -8 - 5 = -13 $$

(c)

$$ -4 - 9 = -13 $$

(d)

$$ -6 - 3 = -9 $$

(a)

$$ 3 - (-9) = 3 + 9 = 12 $$

(b)

$$ 6 - (-5) = 6 + 5 = 11 $$

(c)

$$ -8 - (-2) = -8 + 2 = -6 $$

(d)

$$ -4 - (-7) = -4 + 7 = 3 $$

\begin{align*} \text{Temperature at } A & = -7^\circ \text{C} \\ \text{Temperature at } B & = 16^\circ \text{C} \\ \\ \text{Difference in temperature} & = [16 - (-7)]^\circ \text{C} \\ & = (16 + 7)^\circ \text{C} \\ & = 23^\circ \text{C} \end{align*}

\begin{align*} \text{Temperature in the morning} & = (-5 + 3)^\circ \text{C} \\ & = -2^\circ \text{C} \end{align*}

Calculate each expression to find the number represented by its letter.

\begin{align*} \text{A: } & 7 + (-3) = 7 - 3 = 4 \\ \text{B: } & 19 - (-1) = 19 + 1 = 20 \\ \text{D: } & -5 - (-5) = -5 + 5 = 0 \\ \text{E: } & 8 - (-8) = 8 + 8 = 16 \\ \text{H: } & -14 + 6 = -8 \\ \text{I: } & -2 - (-9) = -2 + 9 = 7 \\ \text{L: } & -1 + (-6) = -7 \\ \text{M: } & 3 + (-12) = 3 - 12 = -9 \\ \text{N: } & 8 - (-2) = 8 + 2 = 10 \\ \text{O: } & -4 + 6 = 2 \\ \text{T: } & -10 - (-5) = -10 + 5 = -5 \\ \text{Y: } & -3 - 3 = -6 \end{align*}

Read the boxes from left to right. For each number above or below a box, write the letter whose calculation gives that number.

$$ \begin{array}{c|rrrrrrr} \text{Number} & -8 & 2 & -7 & 7 & 0 & 4 & -6 \\ \hline \text{Letter} & \text{H} & \text{O} & \text{L} & \text{I} & \text{D} & \text{A} & \text{Y} \end{array} $$

Kumar's favourite day is a HOLIDAY.

(a)

$$ (-14) + (-16) = -30 $$

(b)

$$ -52 + (-27) = -79 $$

(c)

$$ (-39) + (-65) = -104 $$

(d)

$$ -138 + (-22) = -160 $$

(a)

$$ 47 + (-13) = 47 - 13 = 34 $$

(b)

$$ 137 + (-24) = 137 - 24 = 113 $$

(c)

$$ -95 + 113 = 113 - 95 = 18 $$

(d)

$$ -78 + 139 = 139 - 78 = 61 $$

(a)

$$ 16 - 69 = -53 $$

(b)

$$ 14 - 76 = -62 $$

(c)

$$ 88 + (-123) = 88 - 123 = -35 $$

(d)

$$ 76 + (-183) = 76 - 183 = -107 $$

(e)

$$ -73 + 26 = 26 - 73 = -47 $$

(f)

$$ -111 + 12 = 12 - 111 = -99 $$

(a)

$$ -84 - 23 = -107 $$

(b)

$$ -69 - 76 = -145 $$

(c)

$$ -714 - 716 = -1430 $$

(d)

$$ -767 - 697 = -1464 $$

(a)

$$ 24 - (-11) = 24 + 11 = 35 $$

(b)

$$ 38 - (-57) = 38 + 57 = 95 $$

(c)

$$ -69 - (-28) = -69 + 28 = -41 $$

(d)

$$ -34 - (-91) = -34 + 91 = 57 $$

\begin{align*} \text{Altitude of the town} & = -51 \text{ m} \\ \\ \text{Difference in altitude} & = [138 - (-51)] \text{ m} \\ & = (138 + 51) \text{ m} \\ & = 189 \text{ m} \end{align*}

(a)

Negative. The sum of two negative numbers, \(-987\) and \(-654\), is negative.

(b)

Negative. Since \(123\) is smaller than \(456\), subtracting \(456\) from \(123\) gives a negative number.

(c)

Negative. \(436 + (-634) = 436 - 634\). Since \(436\) is smaller than \(634\), the result is negative.

(d)

Positive. \(185 - (-567) = 185 + 567\), which is the sum of two positive numbers.

(i)

$$ \text{Difference} = 3 - (-2) = 3 + 2 = 5 $$

(ii)

The number line includes \(0\) between \(-1\) and \(1\). The BC/AD timeline has no year 0: \(1\) BC is followed immediately by AD \(1\).

(iii)

$$ 2\text{ BC} \rightarrow 1\text{ BC} \rightarrow \text{AD }1 \rightarrow \text{AD }2 \rightarrow \text{AD }3 $$ There are four one-year intervals, so the difference is \(4\) years.

(iv)

One example is a building whose floors are labelled B2, B1, 1, 2, 3, with no floor 0. Floor 1 is the ground floor, and B1 is the basement immediately below it. Moving from B1 to floor 1 is only one storey, just as moving from 1 BC to AD 1 is only one year.

(a)

One pair is \(x = -9,\ y = -1\). $$ (-9) + (-1) = -10 $$

Another pair is \(x = -67,\ y = 57\). $$ (-67) + 57 = -10 $$

(b)

One pair is \(x = 67,\ y = 77\). $$ 67 - 77 = -10 $$

Another pair is \(x = 57,\ y = 67\). $$ 57 - 67 = -10 $$

(a)

$$ 5 \times (-7) = -35 $$

(b)

$$ -8 \times 3 = -24 $$

(c)

$$ (-9)(-4) = (-9) \times (-4) = 36 $$

(d)

$$ -(-716) = 716 $$

(e)

$$ -1 \times (-697) = 697 $$

(f)

$$ -11(-8) = -11 \times (-8) = 88 $$

(g)

$$ -6 \times 0 = 0 $$

(h)

$$ 42(-2) = 42 \times (-2) = -84 $$

(a)

$$ -12 \div 4 = -3 $$

(b)

$$ 16 \div (-2) = -8 $$

(c)

$$ (-18) \div (-9) = 2 $$

(d)

$$ \frac{-14}{7} = -2 $$

(e)

$$ \frac{45}{-5} = -9 $$

(f)

$$ \frac{-18}{-3} = 6 $$

(a)

$$ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 $$

(b)

$$ \pm 1, \pm 23 $$

(c)

$$ \pm 1, \pm 2, \pm 4, \pm 8, \pm 16 $$

(d)

$$ \pm 1 $$

(a)

$$ 10, -10 $$

(b)

$$ 24, -24 $$

(c)

$$ 34, -34 $$

(d)

Not possible. Every multiple of \(0\) is \(0\), which is neither positive nor negative.

(a)

$$ \text{Square roots of } 81 = \pm \sqrt{81} = \pm \sqrt{9 \times 9} = \pm 9 $$

(b)

$$ \text{Square roots of } 16 = \pm \sqrt{16} = \pm \sqrt{4 \times 4} = \pm 4 $$

(c)

$$ \text{Square roots of } 25 = \pm \sqrt{25} = \pm \sqrt{5 \times 5} = \pm 5 $$

(d)

$$ \text{Square roots of } 100 = \pm \sqrt{100} = \pm \sqrt{10 \times 10} = \pm 10 $$

(a)

$$ \sqrt{81} = \sqrt{9 \times 9} = 9 $$

(b)

$$ \sqrt{4} = \sqrt{2 \times 2} = 2 $$

(c)

$$ -\sqrt{9} = -\sqrt{3 \times 3} = -3 $$

(d)

Not possible in the real numbers. The square of any real number is non-negative, so no real number has a square of \(-64\).

(a)

$$ (-4)^3 = (-4) \times (-4) \times (-4) = -64 $$

(b)

$$ -4^3 = -(4 \times 4 \times 4) = -64 $$

(c)

$$ -(-4^3) = -[-(4 \times 4 \times 4)] = -(-64) = 64 $$

(d)

$$ \sqrt[3]{8} = \sqrt[3]{2 \times 2 \times 2} = 2 $$

(e)

$$ -\sqrt[3]{125} = -\sqrt[3]{5 \times 5 \times 5} = -5 $$

(f)

$$ \sqrt[3]{-216} = \sqrt[3]{(-6) \times (-6) \times (-6)} = -6 $$

(g)

$$ -\sqrt[3]{-64} = -\sqrt[3]{(-4) \times (-4) \times (-4)} = -(-4) = 4 $$

(h)

$$ \sqrt[3]{-1000} = \sqrt[3]{(-10) \times (-10) \times (-10)} = -10 $$

(a)

\begin{align*} -55 + (-10) - 12 & = -55 - 10 - 12 \\ & = -65 - 12 \\ & = -77 \end{align*}

(b)

\begin{align*} -12 - [(-8) - (-2)] + 3 & = -12 - (-8 + 2) + 3 \\ & = -12 - (-6) + 3 \\ & = -12 + 6 + 3 \\ & = -6 + 3 \\ & = -3 \end{align*}

(c)

\begin{align*} -100 + (-45) + (-5) + 20 & = -100 - 45 - 5 + 20 \\ & = -145 - 5 + 20 \\ & = -150 + 20 \\ & = -130 \end{align*}

(d)

\begin{align*} -2 + 3 \times 15 & = -2 + 45 \\ & = 43 \end{align*}

(e)

\begin{align*} (-5 - 2)(-3) & = (-7) \times (-3) \\ & = 21 \end{align*}

(f)

\begin{align*} -25 \times (-4) \div (-12 + 32) & = -25 \times (-4) \div 20 \\ & = 100 \div 20 \\ & = 5 \end{align*}

(g)

\begin{align*} 3 \times (-3)^2 - (7 - 2)^2 & = 3 \times (-3)^2 - 5^2 \\ & = 3 \times [(-3) \times (-3)] - 5 \times 5 \\ & = 3 \times 9 - 25 \\ & = 27 - 25 \\ & = 2 \end{align*}

(h)

\begin{align*} 5[3 \times (-2) - 10] & = 5 \times (-6 - 10) \\ & = 5 \times (-16) \\ & = -80 \end{align*}

(i)

\begin{align*} -12 \div [2^2 - (-2)] & = -12 \div [2 \times 2 - (-2)] \\ & = -12 \div (4 + 2) \\ & = -12 \div 6 \\ & = -2 \end{align*}

(j)

\begin{align*} \sqrt{10 - 3 \times (-2)} & = \sqrt{10 - (-6)} \\ & = \sqrt{10 + 6} \\ & = \sqrt{16} \\ & = \sqrt{4 \times 4} \\ & = 4 \end{align*}

Negative. The product \(-987 \times (-654)\) is positive because multiplying two negative numbers gives a positive number. Dividing this positive product by the negative number \(-321\) gives a negative result.

One possible positive factor is \(1\), and one possible negative factor is \(-1\), since $$ 0 = 1 \times 0 = (-1) \times 0. $$

No. The square root symbol gives the non-negative square root, so \(\sqrt{n^2}\) cannot equal \(n\) when \(n\) is negative. For example, let \(n = -3\): $$ \sqrt{(-3)^2} = \sqrt{9} = \sqrt{3 \times 3} = 3 \ne -3. $$ In general, \(\sqrt{n^2} = |n|\), so \(\sqrt{n^2} = n\) holds only when \(n \ge 0\).

Since \(n\) is in the denominator, \(n \ne 0\). For \(\frac{198}{n^2}\) to be an integer, \(n^2\) must be a positive perfect-square factor of \(198\). $$ 198 = 2 \times 3^2 \times 11 $$ A perfect-square factor must have even powers of its prime factors. Thus the only positive perfect-square factors of \(198\) are \(1\) and \(9\).

\begin{align*} n^2 & = 1 \text{ or } 9 \\ n & = \pm \sqrt{1} \text{ or } \pm \sqrt{9} \\ & = \pm \sqrt{1 \times 1} \text{ or } \pm \sqrt{3 \times 3} \\ & = \pm 1 \text{ or } \pm 3 \end{align*}

(a)

\begin{align*} 24 \times (-2) \times 5 \div (-6) & = -48 \times 5 \div (-6) \\ & = -240 \div (-6) \\ & = 40 \end{align*}

(b)

\begin{align*} 4 \times 10 - 13 \times (-5) & = 40 - (-65) \\ & = 40 + 65 \\ & = 105 \end{align*}

(c)

\begin{align*} 160 \div \sqrt[3]{-8} - 20 \div (-5) & = 160 \div \sqrt[3]{(-2) \times (-2) \times (-2)} - 20 \div (-5) \\ & = 160 \div (-2) - 20 \div (-5) \\ & = -80 - (-4) \\ & = -80 + 4 \\ & = -76 \end{align*}

(d)

\begin{align*} \sqrt{5^2 - 3^2} - (57 - 77) \div 2 & = \sqrt{5 \times 5 - 3 \times 3} - (-20) \div 2 \\ & = \sqrt{25 - 9} - (-20) \div 2 \\ & = \sqrt{16} - (-20) \div 2 \\ & = \sqrt{4 \times 4} - (-20) \div 2 \\ & = 4 - (-10) \\ & = 4 + 10 \\ & = 14 \end{align*}

(e)

\begin{align*} [(12 - 18) \div 3 - (-1)] \times (-4)^3 & = [(-6) \div 3 + 1] \times (-4)^3 \\ & = (-2 + 1) \times (-4)^3 \\ & = (-1) \times [(-4) \times (-4) \times (-4)] \\ & = (-1) \times (-64) \\ & = 64 \end{align*}

(f)

\begin{align*} (5 - 2)^3 \times 2 + [-4 + (-7)] \div (-2 + 4)^2 & = 3^3 \times 2 + (-11) \div 2^2 \\ & = (3 \times 3 \times 3) \times 2 + (-11) \div (2 \times 2) \\ & = 27 \times 2 + (-11) \div 4 \\ & = 54 - \frac{11}{4} \\ & = \frac{216}{4} - \frac{11}{4} \\ & = \frac{205}{4} \\ & = 51 \frac{1}{4} \end{align*}

(g)

\begin{align*} \{-10 - [12 + (-3)^2] + 3^3\} \div (-3) & = \{-10 - [12 + (-3) \times (-3)] + 3 \times 3 \times 3\} \div (-3) \\ & = \{-10 - [12 + 9] + 27\} \div (-3) \\ & = (-10 - 21 + 27) \div (-3) \\ & = (-31 + 27) \div (-3) \\ & = (-4) \div (-3) \\ & = \frac{4}{3} \\ & = 1 \frac{1}{3} \end{align*}

(h)

\begin{align*} \sqrt[3]{-2 \times (-37) - [-2(-3) + 8 \times (-2) - 8 \times 2] + 5^2} & = \sqrt[3]{74 - [6 + (-16) - 16] + 5 \times 5} \\ & = \sqrt[3]{74 - [6 - 16 - 16] + 25} \\ & = \sqrt[3]{74 - (-26) + 25} \\ & = \sqrt[3]{74 + 26 + 25} \\ & = \sqrt[3]{125} \\ & = \sqrt[3]{5 \times 5 \times 5} \\ & = 5 \end{align*}

Number of unanswered questions \(= 30 - 17 - 5 = 8\).

\begin{align*} \text{Total score} & = 17 \times 3 + 5 \times (-1) + 8 \times 0 \\ & = 51 - 5 + 0 \\ & = 46 \end{align*}

(i)

(ii)

(a)

\begin{align*} -\frac{1}{2} + \left(-\frac{3}{4}\right) & = -\frac{2}{4} - \frac{3}{4} \\ & = -\frac{5}{4} \\ & = -1 \frac{1}{4} \end{align*}

(b)

\begin{align*} 3 \frac{1}{5} - 5 \frac{1}{2} & = \frac{16}{5} - \frac{11}{2} \\ & = \frac{32}{10} - \frac{55}{10} \\ & = -\frac{23}{10} \\ & = -2 \frac{3}{10} \end{align*}

(c)

\begin{align*} 3 \frac{1}{8} + \left(-\frac{1}{4}\right) & = \frac{25}{8} - \frac{1}{4} \\ & = \frac{25}{8} - \frac{2}{8} \\ & = \frac{23}{8} \\ & = 2 \frac{7}{8} \end{align*}

(d)

\begin{align*} 4 \frac{1}{6} - \left(-4 \frac{2}{3}\right) & = \frac{25}{6} + \frac{14}{3} \\ & = \frac{25}{6} + \frac{28}{6} \\ & = \frac{53}{6} \\ & = 8 \frac{5}{6} \end{align*}

(a)

\begin{align*} 2 \frac{3}{5} \times \frac{15}{26} & = \frac{13}{5} \times \frac{15}{26} \\ & = \frac{3}{2} \\ & = 1 \frac{1}{2} \end{align*}

(b)

\begin{align*} \frac{15}{8} \times \frac{4}{3} & = \frac{5}{2} \\ & = 2 \frac{1}{2} \end{align*}

(c)

\begin{align*} \frac{7}{11} \times 1 \frac{4}{7} & = \frac{7}{11} \times \frac{11}{7} \\ & = 1 \end{align*}

(d)

\begin{align*} 1 \frac{1}{9} \times 2 \frac{5}{8} & = \frac{10}{9} \times \frac{21}{8} \\ & = \frac{5}{3} \times \frac{7}{4} \\ & = \frac{35}{12} \\ & = 2 \frac{11}{12} \end{align*}

(a)

\begin{align*} \frac{15}{4} \div \frac{5}{2} & = \frac{15}{4} \times \frac{2}{5} \\ & = \frac{3}{2} \\ & = 1 \frac{1}{2} \end{align*}

(b)

\begin{align*} 1 \frac{7}{9} \div \frac{4}{3} & = \frac{16}{9} \div \frac{4}{3} \\ & = \frac{16}{9} \times \frac{3}{4} \\ & = \frac{4}{3} \\ & = 1 \frac{1}{3} \end{align*}

(c)

\begin{align*} 9 \frac{2}{9} \div 9 & = \frac{83}{9} \div 9 \\ & = \frac{83}{9} \times \frac{1}{9} \\ & = \frac{83}{81} \\ & = 1 \frac{2}{81} \end{align*}

(d)

\begin{align*} 2 \frac{2}{15} \div 1 \frac{23}{25} & = \frac{32}{15} \div \frac{48}{25} \\ & = \frac{32}{15} \times \frac{25}{48} \\ & = \frac{2}{3} \times \frac{5}{3} \\ & = \frac{10}{9} \\ & = 1 \frac{1}{9} \end{align*}

(a)

\begin{align*} \frac{64}{15} \times \left(-\frac{3}{8}\right) & = \frac{8}{5} \times (-1) \\ & = -\frac{8}{5} \\ & = -1 \frac{3}{5} \end{align*}

(b)

\begin{align*} \frac{4}{15} \div \left(-\frac{10}{3}\right) & = \frac{4}{15} \times \left(-\frac{3}{10}\right) \\ & = \frac{2}{5} \times \left(-\frac{1}{5}\right) \\ & = -\frac{2}{25} \end{align*}

(c)

\begin{align*} -6 \frac{1}{8} \times \frac{3}{14} & = -\frac{49}{8} \times \frac{3}{14} \\ & = -\frac{7}{8} \times \frac{3}{2} \\ & = -\frac{21}{16} \\ & = -1 \frac{5}{16} \end{align*}

(d)

\begin{align*} -2 \frac{1}{2} \times 4 \frac{2}{5} & = -\frac{5}{2} \times \frac{22}{5} \\ & = (-1) \times 11 \\ & = -11 \end{align*}

(e)

\begin{align*} -1 \frac{1}{4} \div \frac{3}{8} & = -\frac{5}{4} \div \frac{3}{8} \\ & = -\frac{5}{4} \times \frac{8}{3} \\ & = -5 \times \frac{2}{3} \\ & = -\frac{10}{3} \\ & = -3 \frac{1}{3} \end{align*}

(f)

\begin{align*} -\frac{8}{9} \div \left(-1 \frac{2}{3}\right) & = -\frac{8}{9} \div \left(-\frac{5}{3}\right) \\ & = -\frac{8}{9} \times \left(-\frac{3}{5}\right) \\ & = \frac{8}{15} \end{align*}

(a)

\begin{align*} (-4.6) + (-3.2) & = -(4.6 + 3.2) \\ & = -7.8 \end{align*}

(b)

\begin{align*} 2.5 - 6.8 & = -(6.8 - 2.5) \\ & = -4.3 \end{align*}

(c)

\begin{align*} 6.9 + (-4.7) & = 6.9 - 4.7 \\ & = 2.2 \end{align*}

(d)

\begin{align*} 12.6 + (-35.7) & = 12.6 - 35.7 \\ & = -(35.7 - 12.6) \\ & = -23.1 \end{align*}

(e)

\begin{align*} -29.3 + 11.2 & = 11.2 - 29.3 \\ & = -(29.3 - 11.2) \\ & = -18.1 \end{align*}

(f)

\begin{align*} -15.3 - 27.9 & = -(15.3 + 27.9) \\ & = -43.2 \end{align*}

(g)

\begin{align*} 11.8 - (-2.3) & = 11.8 + 2.3 \\ & = 14.1 \end{align*}

(h)

\begin{align*} -0.7 - (-1.5) & = -0.7 + 1.5 \\ & = 1.5 - 0.7 \\ & = 0.8 \end{align*}

(a)

\begin{align*} 14.72 \times 1.2 & = \frac{1472}{100} \times \frac{12}{10} \\ & = \frac{1472 \times 12}{1000} \\ & = \frac{17\,664}{1000} \\ & = 17.664 \end{align*}

(b)

\begin{align*} 130.4 \times 0.15 & = \frac{1304}{10} \times \frac{15}{100} \\ & = \frac{1304 \times 15}{1000} \\ & = \frac{19\,560}{1000} \\ & = 19.56 \end{align*}

(c)

\begin{align*} 0.27 \times 0.08 & = \frac{27}{100} \times \frac{8}{100} \\ & = \frac{27 \times 8}{10\,000} \\ & = \frac{216}{10\,000} \\ & = 0.0216 \end{align*}

(d)

\begin{align*} 0.25 \times 1.963 & = \frac{25}{100} \times \frac{1963}{1000} \\ & = \frac{25 \times 1963}{100\,000} \\ & = \frac{49\,075}{100\,000} \\ & = 0.49075 \end{align*}

(a)

\begin{align*} 0.81 \div 0.3 & = \frac{0.81}{0.3} \\ & = \frac{0.81 \times 10}{0.3 \times 10} \\ & = \frac{8.1}{3} \\ & = 2.7 \end{align*}

(b)

\begin{align*} 1.32 \div 0.12 & = \frac{1.32}{0.12} \\ & = \frac{1.32 \times 100}{0.12 \times 100} \\ & = \frac{132}{12} \\ & = 11 \end{align*}

(c)

\begin{align*} 3.426 \div 0.06 & = \frac{3.426}{0.06} \\ & = \frac{3.426 \times 100}{0.06 \times 100} \\ & = \frac{342.6}{6} \\ & = 57.1 \end{align*}

(d)

\begin{align*} 4.35 \div 1.5 & = \frac{4.35}{1.5} \\ & = \frac{4.35 \times 10}{1.5 \times 10} \\ & = \frac{43.5}{15} \\ & = 2.9 \end{align*}

(a)

\begin{align*} 4 \frac{2}{7} - 6 \frac{1}{3} - \left(-\frac{8}{21}\right) & = \frac{30}{7} - \frac{19}{3} + \frac{8}{21} \\ & = \frac{90}{21} - \frac{133}{21} + \frac{8}{21} \\ & = \frac{90 - 133 + 8}{21} \\ & = -\frac{35}{21} \\ & = -\frac{5}{3} \\ & = -1 \frac{2}{3} \end{align*}

(b)

\begin{align*} 2 \frac{1}{5} - \left(-\frac{3}{4}\right) + \left(-7 \frac{1}{10}\right) & = \frac{11}{5} + \frac{3}{4} - \frac{71}{10} \\ & = \frac{44}{20} + \frac{15}{20} - \frac{142}{20} \\ & = \frac{44 + 15 - 142}{20} \\ & = -\frac{83}{20} \\ & = -4 \frac{3}{20} \end{align*}

(c)

\begin{align*} -9 + \left(-2 \frac{1}{8}\right) + \left(-\frac{4}{3}\right) & = -9 - \frac{17}{8} - \frac{4}{3} \\ & = -\frac{216}{24} - \frac{51}{24} - \frac{32}{24} \\ & = \frac{-216 - 51 - 32}{24} \\ & = -\frac{299}{24} \\ & = -12 \frac{11}{24} \end{align*}

(d)

\begin{align*} -\frac{1}{5} + \frac{19}{4} + \left(-\frac{7}{2}\right) & = -\frac{1}{5} + \frac{19}{4} - \frac{7}{2} \\ & = -\frac{4}{20} + \frac{95}{20} - \frac{70}{20} \\ & = \frac{-4 + 95 - 70}{20} \\ & = \frac{21}{20} \\ & = 1 \frac{1}{20} \end{align*}

(e)

\begin{align*} -\frac{8}{3} - \frac{1}{6} - \left(-2 \frac{1}{8}\right) & = -\frac{8}{3} - \frac{1}{6} + \frac{17}{8} \\ & = -\frac{64}{24} - \frac{4}{24} + \frac{51}{24} \\ & = \frac{-64 - 4 + 51}{24} \\ & = -\frac{17}{24} \end{align*}

(f)

\begin{align*} 5 \frac{3}{4} - 2 \frac{5}{6} + \left(-\frac{23}{15}\right) - \left(-4 \frac{7}{10}\right) & = \frac{23}{4} - \frac{17}{6} - \frac{23}{15} + \frac{47}{10} \\ & = \frac{345}{60} - \frac{170}{60} - \frac{92}{60} + \frac{282}{60} \\ & = \frac{345 - 170 - 92 + 282}{60} \\ & = \frac{365}{60} \\ & = \frac{73}{12} \\ & = 6 \frac{1}{12} \end{align*}

\begin{align*} \text{Time spent visiting the homes} & = 8 \frac{1}{16} \times \frac{4}{7} \text{ h} \\ & = \frac{129}{16} \times \frac{4}{7} \text{ h} \\ & = \frac{129}{4} \times \frac{1}{7} \text{ h} \\ & = \frac{129}{28} \text{ h} \\ & = 4 \frac{17}{28} \text{ h} \end{align*}

(a)

\begin{align*} -\frac{5}{7} \times \left(-\frac{28}{15} + 1 \frac{2}{3}\right) & = -\frac{5}{7} \times \left(-\frac{28}{15} + \frac{5}{3}\right) \\ & = -\frac{5}{7} \times \left(-\frac{28}{15} + \frac{25}{15}\right) \\ & = -\frac{5}{7} \times \left(-\frac{3}{15}\right) \\ & = -\frac{5}{7} \times \left(-\frac{1}{5}\right) \\ & = \frac{1}{7} \end{align*}

(b)

\begin{align*} \left[-\left(\frac{1}{2}\right)^2 - \left(-\frac{1}{3}\right)\right] \div \left(\frac{1}{4} - \frac{1}{3}\right) & = \left(-\frac{1}{4} + \frac{1}{3}\right) \div \left(\frac{1}{4} - \frac{1}{3}\right) \\ & = \left(-\frac{3}{12} + \frac{4}{12}\right) \div \left(\frac{3}{12} - \frac{4}{12}\right) \\ & = \frac{1}{12} \div \left(-\frac{1}{12}\right) \\ & = \frac{1}{12} \times (-12) \\ & = -1 \end{align*}

(c)

\begin{align*} 10 - \frac{15}{8} \times \left(\frac{3}{2} \div 4 \frac{1}{2}\right) + \left(-\frac{1}{4}\right) & = 10 - \frac{15}{8} \times \left(\frac{3}{2} \div \frac{9}{2}\right) - \frac{1}{4} \\ & = 10 - \frac{15}{8} \times \left(\frac{3}{2} \times \frac{2}{9}\right) - \frac{1}{4} \\ & = 10 - \frac{15}{8} \times \frac{1}{3} - \frac{1}{4} \\ & = 10 - \frac{5}{8} - \frac{2}{8} \\ & = 10 - \frac{7}{8} \\ & = 9 \frac{1}{8} \end{align*}

(d)

\begin{align*} \left(-\frac{3}{2}\right)^2 \times \left(\frac{1}{15} - 2 \frac{1}{3}\right) & = \left(-\frac{3}{2}\right) \times \left(-\frac{3}{2}\right) \times \left(\frac{1}{15} - \frac{7}{3}\right) \\ & = \frac{9}{4} \times \left(\frac{1}{15} - \frac{35}{15}\right) \\ & = \frac{9}{4} \times \left(-\frac{34}{15}\right) \\ & = -\frac{51}{10} \\ & = -5 \frac{1}{10} \end{align*}

(a)

\begin{align*} \left(\frac{1}{2}\right)^3 - \left(2 \frac{3}{4}\right)^2 + \left(-\frac{3}{4}\right) & = \left(\frac{1}{2}\right)^3 - \left(\frac{11}{4}\right)^2 - \frac{3}{4} \\ & = \frac{1}{8} - \frac{121}{16} - \frac{3}{4} \\ & = \frac{2}{16} - \frac{121}{16} - \frac{12}{16} \\ & = -\frac{131}{16} \\ & = -8 \frac{3}{16} \end{align*}

(b)

\begin{align*} \frac{7}{3} + \frac{4}{9} \times \left(-\frac{1}{2}\right)^2 & = \frac{7}{3} + \frac{4}{9} \times \frac{1}{4} \\ & = \frac{7}{3} + \frac{1}{9} \\ & = \frac{21}{9} + \frac{1}{9} \\ & = \frac{22}{9} \\ & = 2 \frac{4}{9} \end{align*}

(c)

\begin{align*} -\frac{11}{4} - \sqrt[3]{2 - 5 \frac{3}{8}} & = -\frac{11}{4} - \sqrt[3]{2 - \frac{43}{8}} \\ & = -\frac{11}{4} - \sqrt[3]{-\frac{27}{8}} \\ & = -\frac{11}{4} - \left(-\frac{3}{2}\right) \\ & = -\frac{11}{4} + \frac{6}{4} \\ & = -\frac{5}{4} \\ & = -1 \frac{1}{4} \end{align*}

(d)

\begin{align*} \sqrt{\frac{11}{12} \div \left(\frac{5}{2} - \frac{5}{6} \times \frac{19}{10}\right)} & = \sqrt{\frac{11}{12} \div \left(\frac{5}{2} - \frac{19}{12}\right)} \\ & = \sqrt{\frac{11}{12} \div \left(\frac{30}{12} - \frac{19}{12}\right)} \\ & = \sqrt{\frac{11}{12} \div \frac{11}{12}} \\ & = \sqrt{1} \\ & = 1 \end{align*}

(a)

\begin{align*} -88.8 - 16.24 & = -(88.80 + 16.24) \\ & = -105.04 \end{align*}

(b)

\begin{align*} 4.73 - 14 & = -(14.00 - 4.73) \\ & = -9.27 \end{align*}

(c)

\begin{align*} 7.513 + (-18.9) & = 7.513 - 18.900 \\ & = -(18.900 - 7.513) \\ & = -11.387 \end{align*}

(d)

\begin{align*} -23.6 + 18.251 & = 18.251 - 23.600 \\ & = -(23.600 - 18.251) \\ & = -5.349 \end{align*}

(e)

\begin{align*} 123.4 - (-5.67) & = 123.40 + 5.67 \\ & = 129.07 \end{align*}

(f)

\begin{align*} -53.9 - (-32.17) & = -53.90 + 32.17 \\ & = -(53.90 - 32.17) \\ & = -21.73 \end{align*}

(g)

\begin{align*} 73.8 - (-5.79) + (-16.732) & = 73.800 + 5.790 - 16.732 \\ & = 79.590 - 16.732 \\ & = 62.858 \end{align*}

(h)

\begin{align*} -18.913 - (-2.78) - 7.8 & = -18.913 + 2.780 - 7.800 \\ & = -16.133 - 7.800 \\ & = -23.933 \end{align*}

(a)

\begin{align*} \frac{0.15}{0.5} \times \left(\frac{-0.16}{1.2}\right) & = \frac{1.5}{5} \times \left(\frac{-1.6}{12}\right) \\ & = 0.3 \times \left(\frac{-1.6}{12}\right) \\ & = \frac{0.3 \times (-1.6)}{12} \\ & = \frac{-0.48}{12} \\ & = -0.04 \end{align*}

(b)

\begin{align*} \frac{0.027}{0.03} \times \left(\frac{1.4}{-0.18}\right) & = \frac{2.7}{3} \times \left(\frac{140}{-18}\right) \\ & = 0.9 \times \left(\frac{140}{-18}\right) \\ & = \frac{0.9 \times 140}{-18} \\ & = \frac{126}{-18} \\ & = -7 \end{align*}

(c)

\begin{align*} -0.4^2 \times (-1.3) \div 0.8 - 0.62 & = -(0.4 \times 0.4) \times (-1.3) \div 0.8 - 0.62 \\ & = -0.16 \times (-1.3) \div 0.8 - 0.62 \\ & = 0.208 \div 0.8 - 0.62 \\ & = 0.26 - 0.62 \\ & = -0.36 \end{align*}

(d)

\begin{align*} (-0.2)^3 \times \frac{27}{1.6} + 0.105 & = [(-0.2) \times (-0.2) \times (-0.2)] \times \frac{27}{1.6} + 0.105 \\ & = -0.008 \times \frac{27}{1.6} + 0.105 \\ & = \frac{-0.216}{1.6} + 0.105 \\ & = -0.135 + 0.105 \\ & = -0.03 \end{align*}

(e)

\begin{align*} -2.4 - (-1.6)^2 \div (-0.8)^3 & = -2.4 - [(-1.6) \times (-1.6)] \div [(-0.8) \times (-0.8) \times (-0.8)] \\ & = -2.4 - 2.56 \div (-0.512) \\ & = -2.4 - (-5) \\ & = -2.4 + 5 \\ & = 2.6 \end{align*}

(f)

\begin{align*} [-7.54 + (-5.79)] \times [7 \div (-4)] & = (-13.33) \times (-1.75) \\ & = 13.33 \times 1.75 \\ & = 23.3275 \end{align*}

(a)

\begin{align*} \left(\frac{\pi + 5 \frac{1}{2}}{-2.1}\right)^2 & = \left(\frac{\pi + 5.5}{-2.1}\right)^2 \\ & \approx 16.9335881158 \\ & \approx 16.934 \text{ (3 d.p.)} \end{align*}

(b)

\begin{align*} -\frac{\pi^2 + \sqrt{2}}{7 - \sqrt[3]{4}} & \approx -2.0847319507 \\ & \approx -2.085 \text{ (3 d.p.)} \end{align*}

(c)

\begin{align*} \frac{\sqrt[3]{14^2 + 19^2}}{\pi - 4.55} & = \frac{\sqrt[3]{557}}{\pi - 4.55} \\ & \approx -5.8419358450 \\ & \approx -5.842 \text{ (3 d.p.)} \end{align*}

(d)

\begin{align*} \sqrt{\frac{4.6^2 + 8.3^2 - \left(6 \frac{1}{2}\right)^2}{2 \times 4.6 - 8.3}} & = \sqrt{\frac{21.16 + 68.89 - 42.25}{9.2 - 8.3}} \\ & = \sqrt{\frac{47.8}{0.9}} \\ & \approx 7.2877370364 \\ & \approx 7.288 \text{ (3 d.p.)} \end{align*}

A rational number must be expressible as \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \ne 0\). Since \(C\) and \(d\) may not be integers, writing \(\pi = \frac{C}{d}\) does not prove that \(\pi\) is rational.

\begin{align*} \text{Let } x & = 2. \dot{9} \\ \\ 10x & = 2.\dot{9} \times 10 \\ 10x & = 2.999 \ldots \times 10 \\ 10x & = 29.999 \ldots \\ 10x & = 29.\dot{9} \\ \\ 10x - x & = 29.\dot{9} - 2.\dot{9} \\ 9x & = 27 \\ x & = 27 \div 9 \\ x & = 3 \\ \\ \therefore 3 - 2.\dot{9} & = 3 - x \\ & = 3 - 3 \\ & = 0 \end{align*}

(a)

\begin{align*} -7 - 38 & = -45 \\ 8 + (-55) & = -47 \end{align*} $$ \therefore -7 - 38 \gt 8 + (-55) $$

(b)

\begin{align*} 2.36 - 10.58 & = -8.22 \\ -11.97 - (-2.69) & = -11.97 + 2.69 = -9.28 \end{align*} $$ \therefore 2.36 - 10.58 \gt -11.97 - (-2.69) $$

(c)

\begin{align*} -5 \times 1.5 & = -7.5 \\ 50 \div (-8) & = -6.25 \end{align*} $$ \therefore -5 \times 1.5 \lt 50 \div (-8) $$

(d)

\begin{align*} 7 \frac{1}{5} - \left(-3 \frac{3}{10}\right) & = 7.2 + 3.3 = 10.5 \\ 19 \frac{2}{5} + \left(-8 \frac{1}{10}\right) & = 19.4 - 8.1 = 11.3 \end{align*} $$ \therefore 7 \frac{1}{5} - \left(-3 \frac{3}{10}\right) \lt 19 \frac{2}{5} + \left(-8 \frac{1}{10}\right) $$

(a)

(b)

(a)

\begin{align*} 13 - (-54) + (-9) & = 13 + 54 - 9 \\ & = 67 - 9 \\ & = 58 \end{align*}

(b)

\begin{align*} -74 + (-46) - 120 & = -74 - 46 - 120 \\ & = -120 - 120 \\ & = -240 \end{align*}

(c)

\begin{align*} (-3 - 5) \times (-3 - 4) & = (-8) \times (-7) \\ & = 56 \end{align*}

(d)

\begin{align*} 4(-5)(-6) \div (-15) & = 4 \times (-5) \times (-6) \div (-15) \\ & = -20 \times (-6) \div (-15) \\ & = 120 \div (-15) \\ & = -8 \end{align*}

(e)

\begin{align*} (-2)^2 - (-2) \times 3 + 2 \times 3^2 & = (-2) \times (-2) - (-6) + 2 \times (3 \times 3) \\ & = 4 + 6 + 18 \\ & = 28 \end{align*}

(f)

\begin{align*} 4 \times 3^2 \div (-6) - (-1)^3 \times (-3)^2 & = 4 \times 9 \div (-6) - (-1) \times 9 \\ & = 36 \div (-6) - (-9) \\ & = -6 + 9 \\ & = 3 \end{align*}

(g)

\begin{align*} -2 \times (-2)^3 \times (-2) \times 3 + (-2) \times 3 \times (-1)^2 & = -2 \times (-8) \times (-2) \times 3 + (-2) \times 3 \times 1 \\ & = 16 \times (-2) \times 3 - 6 \\ & = -32 \times 3 - 6 \\ & = -96 - 6 \\ & = -102 \end{align*}

(h)

\begin{align*} \sqrt[3]{15 + \{12 \times [(-5) \times 7] \div 3\}} & = \sqrt[3]{15 + [12 \times (-35) \div 3]} \\ & = \sqrt[3]{15 + (-420) \div 3} \\ & = \sqrt[3]{15 - 140} \\ & = \sqrt[3]{-125} \\ & = -5 \end{align*}

(a)

\begin{align*} 3 \frac{4}{7} + 1 \frac{2}{5} - \left(-\frac{3}{7}\right) & = \frac{25}{7} + \frac{7}{5} + \frac{3}{7} \\ & = \frac{28}{7} + \frac{7}{5} \\ & = 4 + \frac{7}{5} \\ & = \frac{27}{5} \\ & = 5 \frac{2}{5} \end{align*}

(b)

\begin{align*} \frac{2}{3} - \left(-3 \frac{3}{20}\right) + \left(-\frac{2}{5}\right) & = \frac{2}{3} + \frac{63}{20} - \frac{2}{5} \\ & = \frac{40}{60} + \frac{189}{60} - \frac{24}{60} \\ & = \frac{205}{60} \\ & = \frac{41}{12} \\ & = 3 \frac{5}{12} \end{align*}

(c)

\begin{align*} -6 \frac{4}{9} - 3 \frac{3}{4} - 3 \frac{5}{9} & = -\frac{58}{9} - \frac{15}{4} - \frac{32}{9} \\ & = -\frac{90}{9} - \frac{15}{4} \\ & = -10 - \frac{15}{4} \\ & = -\frac{55}{4} \\ & = -13 \frac{3}{4} \end{align*}

(d)

\begin{align*} \left(-\frac{1}{2} + \frac{1}{3}\right)^2 - \left[\frac{1}{4} + \left(-\frac{1}{3}\right)\right] + \left(-\frac{1}{20}\right) & = \left(-\frac{1}{6}\right)^2 - \left(\frac{1}{4} - \frac{1}{3}\right) - \frac{1}{20} \\ & = \frac{1}{36} - \left(-\frac{1}{12}\right) - \frac{1}{20} \\ & = \frac{1}{36} + \frac{1}{12} - \frac{1}{20} \\ & = \frac{5}{180} + \frac{15}{180} - \frac{9}{180} \\ & = \frac{11}{180} \end{align*}

(e)

\begin{align*} -3 \frac{1}{4} \times \frac{8}{5} \times \left(-1 \frac{2}{13}\right) & = -\frac{13}{4} \times \frac{8}{5} \times \left(-\frac{15}{13}\right) \\ & = -\frac{26}{5} \times \left(-\frac{15}{13}\right) \\ & = 6 \end{align*}

(f)

\begin{align*} \frac{3}{5} \times \left(-\frac{1}{4} - \frac{1}{6}\right) \div \left(-2 \frac{1}{3} + 1 \frac{1}{4}\right) & = \frac{3}{5} \times \left(-\frac{5}{12}\right) \div \left(-\frac{7}{3} + \frac{5}{4}\right) \\ & = \frac{3}{5} \times \left(-\frac{5}{12}\right) \div \left(-\frac{28}{12} + \frac{15}{12}\right) \\ & = -\frac{1}{4} \div \left(-\frac{13}{12}\right) \\ & = -\frac{1}{4} \times \left(-\frac{12}{13}\right) \\ & = \frac{3}{13} \end{align*}

(g)

\begin{align*} -\frac{57}{16} \div 1 \frac{3}{16} - \frac{1}{3} \times \left(-1 \frac{3}{4}\right) & = -\frac{57}{16} \div \frac{19}{16} - \frac{1}{3} \times \left(-\frac{7}{4}\right) \\ & = -\frac{57}{16} \times \frac{16}{19} + \frac{7}{12} \\ & = -3 + \frac{7}{12} \\ & = -\frac{36}{12} + \frac{7}{12} \\ & = -\frac{29}{12} \\ & = -2 \frac{5}{12} \end{align*}

(h)

\begin{align*} -12 \frac{1}{2} + 1 \frac{2}{3} \div (-4) - \left(-\frac{5}{7}\right) \times \left(-1 \frac{1}{20}\right) & = -\frac{25}{2} + \frac{5}{3} \div (-4) - \left(-\frac{5}{7}\right) \times \left(-\frac{21}{20}\right) \\ & = -\frac{25}{2} - \frac{5}{12} - \frac{3}{4} \\ & = -\frac{150}{12} - \frac{5}{12} - \frac{9}{12} \\ & = -\frac{164}{12} \\ & = -\frac{41}{3} \\ & = -13 \frac{2}{3} \end{align*}

(a)

\begin{align*} -12.8 + (-88.2) & = -(12.8 + 88.2) \\ & = -101 \end{align*}

(b)

\begin{align*} 500.3 - (-200.2) - 210.1 & = 500.3 + 200.2 - 210.1 \\ & = 700.5 - 210.1 \\ & = 490.4 \end{align*}

(c)

\begin{align*} 1.44 \div 1.2 \times (-0.4) & = 1.2 \times (-0.4) \\ & = -0.48 \end{align*}

(d)

\begin{align*} (-0.3)^2 \div (-0.2) + (-2.56) & = [(-0.3) \times (-0.3)] \div (-0.2) - 2.56 \\ & = 0.09 \div (-0.2) - 2.56 \\ & = -0.45 - 2.56 \\ & = -3.01 \end{align*}

(a)

\begin{align*} \frac{-18 - [\sqrt[3]{-3375} - (-6)^2]}{\sqrt{4} + 9} & = \frac{-18 - [-15 - 36]}{2 + 9} \\ & = \frac{-18 - (-51)}{11} \\ & = \frac{33}{11} \\ & = 3 \end{align*}

(b)

(a)

(b)

\begin{align*} \text{Let } x & = 1.\dot{6} \\ 10x & = 16.\dot{6} \\ 10x - x & = 16.\dot{6} - 1.\dot{6} \\ 9x & = 15 \\ x & = \frac{15}{9} = \frac{5}{3} \\ \\ \therefore 5 + 1.\dot{6} & = 5 + \frac{5}{3} \\ & = \frac{20}{3} \\ & = 6 \frac{2}{3} \\ & = 6.\dot{6} \end{align*}

(a)

Yes. Since \((-9)^3 = -729\), $$ \sqrt[3]{-729} = -9. $$ The square of every real number is non-negative, so no real number has a square of \(-729\). Hence, \(-\sqrt{-729}\) has no real value.

(b)

Yes. For any integer \(n\), $$ (-n)^3 = (-n) \times (-n) \times (-n) = -n^3. $$ Thus, \(-n\) has a cube of \(-n^3\). Since every real number has exactly one cube root, $$ \sqrt[3]{-n^3} = \sqrt[3]{(-n)^3} = -n. $$

(i)

Two examples are \(p = 67,\ q = 74\) and \(p = 60,\ q = 67\), since \begin{align*} 67 - 74 & = -7 \\ 60 - 67 & = -7 \end{align*}

(ii)

Two examples are \(-0.18\) and \(-0.18\), and \(-0.67\) and \(0.31\), since \begin{align*} -0.18 + (-0.18) & = -0.36 \\ -0.67 + 0.31 & = -0.36 \end{align*}

(i)

(ii)