Answers and Solutions

1. 1.  \[\begin{align*} 4 \times 2^3 + \left(8-3^2\right) &= 4 \times 2^3 + \left(8-9\right) \\[1ex] &= 4 \times 2^3 + (-1) \\[1ex] &= 4 \times 8 + (-1) \\[1ex] &= 32 + (-1) \\[1ex] &= 32 - 1 \\[1ex] &= 31 \end{align*}\]
   2.  \[\begin{align*} \left( \dfrac{1}{2} + \dfrac{1}{4} \right)^2 - \dfrac{3}{8} &= \left( \dfrac{2}{4} + \dfrac{1}{4} \right)^2 - \dfrac{3}{8} \\[1ex] &= \left( \dfrac{3}{4} \right)^2 - \dfrac{3}{8} \\[1ex] &= \dfrac{9}{16} - \dfrac{3}{8} \\[1ex] &= \dfrac{9}{16} - \dfrac{6}{16} \\[1ex] &= \dfrac{3}{16} \\[1ex] \end{align*} \]
   3.  \[ \begin{align*} 5.488 - \left( 0.2^3 + 1.26 \right) &= 5.488 - \left( 0.008 + 1.26 \right) \\[1ex] &= 5.488 - 1.268 \\[1ex] &= 4.22 \end{align*} \]
2. Insert the brackets into the equation in different ways to determine the different answers that are possible. There are a total of \(7\) different answers that can be obtained. An example of how to place brackets to obtain each answer is provided.\[\begin{align*} \left( 2.2 \times 3^2 + 8 \right) \times 6.1 - 6 &= \left( 2.2 \times 9 + 8\right) \times 6.1 - 6 \\[1ex] &= \left(19.8 + 8 \right) \times 6.1 - 6 \\[1ex] &= 27.8 \times 6.1 - 6 \\[1ex] &= 169.58 - 6 \\[1ex] &= 163.58 \end{align*}\]\[\begin{align*} 2.2 \times \left( 3^2 + 8 \right) \times 6.1 - 6 &= 2.2 \times \left( 9 + 8\right) \times 6.1 - 6 \\[1ex] &= 2.2 \times 17 \times 6.1 - 6 \\[1ex] &= 37.4 \times 6.1 - 6 \\[1ex] &= 228.14 - 6 \\[1ex] &= 222.14 \end{align*}\]\[\begin{align*} 2.2 \times 3^2 + \left( 8 \times 6.1 - 6 \right) &= 2.2 \times 9 + \left( 8 \times 6.1 - 6 \right) \\[1ex] &= 2.2 \times 9 + \left(48.8 - 6 \right) \\[1ex] &= 2.2 \times 9 + 42.8 \\[1ex] &= 19.8 + 42.8 \\[1ex] &= 62.6 \end{align*}\]

   Note: The placement of the brackets with \((8 \times 6.1 - 6)\) did not actually affect the answer in that the order of operations was unchanged. Instead of placing the brackets there, brackets could have been inserted around \((8 \times 6.1)\) or \((2.2 \times 3^2)\) without giving an answer different from \(62.6\). 
    

   \[\begin{align*} 2.2 \times 3^2 + 8 \times \left( 6.1 - 6 \right) &= 2.2 \times 9 + 8 \times \left( 6.1 - 6 \right) \\[1ex] &= 2.2 \times 9 + 8 \times 0.1 \\[1ex] &= 19.8 + 8 \times 0.1 \\[1ex] &= 19.8 + 0.8 \\[1ex] &= 20.6 \end{align*}\]\[\begin{align*} 2.2 \times \left( 3^2 + 8 \times 6.1 \right) - 6 &= 2.2 \times \left( 9 + 8\times 6.1 \right) - 6 \\[1ex] &= 2.2 \times \left( 9 + 48.8 \right) - 6 \\[1ex] &= 2.2 \times 57.8 - 6 \\[1ex] &= 127.16 - 6 \\[1ex] &= 121.16 \end{align*}\]\[\begin{align*} 2.2 \times \left( 3^2 + 8 \times 6.1 - 6 \right) &= 2.2 \times \left( 9 + 8\times 6.1 - 6 \right) \\[1ex] &= 2.2 \times \left( 9 + 48.8 - 6 \right) \\[1ex] &= 2.2 \times \left( 57.8 - 6 \right) \\[1ex] &= 2.2 \times 51.8 \\[1ex] &= 113.96 \end{align*}\]\[\begin{align*} \left( 2.2 \times 3\right)^2 + 8 \times 6.1 - 6 &= 6.6^2 + 8 \times 6.1 - 6 \\[1ex] &= 43.56 + 8 \times 6.1 - 6 \\[1ex] &= 43.56 + 48.8 - 6 \\[1ex] &= 92.36 - 6 \\[1ex] &= 86.36 \end{align*}\]
3. 1. \(9.50 \times 4\dfrac{1}{2} + 12.25 \times 4\)
   2. Evaluating, we have\[ \begin{align*} 9.50 \times 4\dfrac{1}{2} + 12.25 \times 4 &= 9.5 \times 4.5 + 12.25 \times 4 \\[1ex] &= 42.75 + 12.25 \times 4 \\[1ex] &= 42.75 + 49 \\[1ex] &= 91.75 \end{align*}\] Therefore, Jenna earned a total of \($91.75\) last weekend.
4. 1. positive; the expression \((-2)-3^2\) is negative, and the product of two negative integers is positive.
   2. \(0\); the expression \(25-5^2\) is equal to \(0\), and the product of \(0\) and any number is \(0\).
   3. negative; the expression \(19+11^2\) is positive and the expression \(14-20\) is negative which, when multiplied together, will give a negative result.
   4. positive; the square of any non-zero integer is positive.
   5. negative; the value of \(135\) is less than the value of \(211^2\) giving a negative result.
5. The expressions ordered from least to greatest are:\[0.8^2 - 2^2, \qquad (0.8-2)^2, \qquad 0.8^2 + 2^2, \qquad (0.8 + 2)^2\]
6. Answers may vary.
   1. The following table provides an example of an expression containing five \(2\)s that is equal to each number between \(1\) and \(9\).

      Number	Expression
      \(1\)	\(\dfrac{2+2+2}{2} -2\)
      \(2\)	\(2\times 2^2 \div (2\times 2)\)
      \(3\)	\(2^2-2+\dfrac{2}{2}\)
      \(4\)	\(\dfrac{2^2+2^2}{2}\)
      \(5\)	\(\dfrac{2+2+2}{2} + 2\)
      \(6\)	\(\dfrac{2}{2} \times 2 + 2+2\)
      \(7\)	\(\dfrac{(2+2)^2 -2}{2}\)
      \(8\)	\(2^2 \times 2^2 \div 2\)
      \(9\)	\(\left( \dfrac{2+2+2}{2} \right)^2\)

   2. The following table provides an example of an expression containing five \(3\)s that is equal to each number between \(1\) and \(9\).

      Number	Expression
      \(1\)	\(\dfrac{3}{3} + (3-3)^3\)
      \(2\)	\(\dfrac{3+3}{3} \times \dfrac{3}{3}\)
      \(3\)	\(\dfrac{3\times 3 \times 3}{3\times 3}\)
      \(4\)	\(\dfrac{3\times 3}{3 \times 3} + 3\)
      \(5\)	\(3+3 - \left(\dfrac{3}{3}\right)^3\)
      \(6\)	\(3 \times 3 - \dfrac{3\times 3}{3}\)
      \(7\)	\(3\times 3 - \dfrac{3+3}{3}\)
      \(8\)	\(\dfrac{3+3}{3} + 3 + 3\)
      \(9\)	\(3^3 \div 3 \times \dfrac{3}{3}\)

7. 1. \((2\times 4 - 3)^2\)
   2. To find the input, work through the machine backwards performing the opposite operations.
      That is, take the square root of \(121\), add \(3\), and divide by \(2\).
      
      Therefore, if \(121\) is the output, then the input was \(7\). To check your answer, start with an input of \(7\) and make sure you produce an output of \(121\).
   3. \(\dfrac{\sqrt{m} + 3}{2}\)
8. 1. Answers may vary.
      \(a=\dfrac{1}{2}\) and \(b=\dfrac{1}{6}\)
   2. Answers may vary, any fraction less than \(-3\) is valid.
      \(c=-\dfrac{7}{2}\)
9. 1. 1. \((4+1)^2 = 25\) and \(4^2+1^2=17\); the difference between these two values is \(8\).
      2. \((2+3)^2=25\) and \(2^2+3^2=13\); the difference between these two values is \(12\).
      3. \((7+3)^2=100\) and \(7^2 + 3^2 = 58\); the difference between these two values is \(42\).
   2. We notice in part a), that the difference between the two values is always \(2\) times the product of the integers being squared.  For example, \(8=2\times 4 \times 1\), \( 12=2\times 2 \times 3\), and \(42=2 \times 7 \times 3\).
      Therefore,\[(a+b)^2 - (a^2 + b^2) = 2 \times a \times b = 2ab\]
   3. A good place to start is by trying a few examples containing numbers instead of variables.
      • \((4-1)^2 = 9\) and \(4^2+1^2=17\); the difference between these two values is \(-8\).
      • \((2-3)^2=1\) and \(2^2+3^2=13\); the difference between these two values is \(-12\).
      • \((7-3)^2=16\) and \(7^2 + 3^2 = 58\); the difference between these two values is \(-42\).
      Notice that the difference between the two values is always \(-2\)  times the product of the two integers you start with. Therefore,\[(a-b)^2 - (a^2 + b^2) = (-2) \times a \times b = -2ab\]