Exercises

1. Evaluate.

   1. \(4 \times 2^3 + \left(8-3^2\right)\)

   2. \(\left( \dfrac{1}{2} + \dfrac{1}{4}\right)^2 - \dfrac{3}{8}\)

   3. \(5.488 - \left(0.2^3 + 1.26\right)\)

2. What different answers are possible when you insert one set of brackets into this expression?

   \[2.2 ~\times ~3~^2 ~+ ~8 ~\times ~6.1 ~- ~6\]
3. Jenna has two part-time jobs.  She earns \($9.50\) per hour babysitting for her neighbour and \($12.25\) per hour working at the Farmers' Market.  Last weekend, she worked \(4\dfrac{1}{2}\) hours babysitting and \(4\) hours at the market.
   1. Write an expression to represent Jenna's total earnings from last weekend.
   2. Evaluate your expression to calculate her total earnings.
4. Without calculating the final answer, determine whether each answer is positive, negative, or \(0\).
   1. \(\Big( (-2) - 3^2 \Big) \times (-7)\)
   2. \(\left(25 - 5^2\right) \times 450\)
   3. \(\left( 19 + 11^2\right) \times (14 - 20)\)
   4. \((135-211)^2\)
   5. \(135 - 211^2\)
5. Order the following expressions from least to greatest.\[(0.8 + 2)^2, \qquad 0.8^2 + 2^2, \qquad (0.8-2)^2, \qquad 0.8^2 - 2^2\]
6. Here are two ways of using five \(2\)s and the order of operations to express the number \(0\).
   \(\left(\dfrac{2}{2} - \dfrac{2}{2}\right)\div 2 \quad\quad\) and \(\quad\quad \left(2^2-2^2\right) \div 2\)
   1. Use five \(2\)s, the operations \(\times,~ \div, ~+,~ -\), brackets, and/or exponents to express each number from \(1\) to \(9\).
   2. Express each number from \(1\) to \(9\) using five \(3\)s and the order of operations.
7. The following machine doubles each input, subtracts \(3\), and then squares the result.
   
   For example, if we input \(5\), then double \(5\) to get \(10\), subtract \(3\) to get \(7\), and finally square \(7\) to get \(49\).
   1. Write the expression that comes out of the machine if the input is \(4\).
   2. If \(121\) is the output, what was the input?
   3. If the machine outputs a positive integer, \(m\), write an expression for the input.
8. The variables \(a,~b,\) and \(c\) represent three different fractions. Determine values for \(a,~b,\) and \(c\) that make each statement true.
   For example, in the equation \(\dfrac{2}{3} \times a = \dfrac{8}{15}\), the value \(a=\dfrac{4}{5}\) makes the equation true.
   1. \(\dfrac{1}{3} \times a + b \lt \dfrac{1}{3} + a \times b\)
   2. \(c + \dfrac{3}{4} \lt c \times \dfrac{3}{4}\)
9. 1. Compare the values for each of the following. What do you notice? 
      1. \((4+1)^2\) and \(4^2 + 1^2\)
      2. \((2+3)^2\) and \(2^2 + 3^2\)
      3. \((7+3)^2\) and \(7^2 + 3^2\)
   2. Using your examples from part a), predict the difference between \(\left(a+b\right)^2\) and \(\left(a^2+b^2\right)\), where \(a\) and \(b\) represent positive integers.
   3. Predict the difference between  \(\left(a-b\right)^2\) and \(\left(a^2+b^2\right)\), where \(a\) and \(b\) represent positive integers.