Using the Order of Operations

Use the order of operations to evaluate each of the following expressions.

ⓐ ${(3 \cdot 2)}^{2} - 4 (6 + 2)$
ⓑ $\frac{5^{2} - 4}{7} - \sqrt{11 - 2}$
ⓒ $6 - | 5 - 8 | + 3 (4 - 1)$
ⓓ $\frac{14 - 3 \cdot 2}{2 \cdot 5 - 3^{2}}$
ⓔ $7 (5 \cdot 3) - 2 [(6 - 3) - 4^{2}] + 1$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
1. ⓐ
  $\begin{matrix} {(3 \cdot 2)}^{2} - 4 (6 + 2) & = & {(6)}^{2} - 4 (8) & Simplify parentheses. \\ = & 36 - 4 (8) & Simplify exponent. \\ = & 36 - 32 & Simplify multiplication. \\ = & 4 & Simplify subtraction. \end{matrix}$
2. ⓑ
  $\begin{matrix} \frac{5^{2} - 4}{7} - \sqrt{11 - 2} & = & \frac{5^{2} - 4}{7} - \sqrt{9} & Simplify grouping symbols (radical). \\ = & \frac{5^{2} - 4}{7} - 3 & Simplify radical. \\ = & \frac{25 - 4}{7} - 3 & Simplify exponent. \\ = & \frac{21}{7} - 3 & Simplify subtraction in numerator. \\ = & 3 - 3 & Simplify division. \\ = & 0 & Simplify subtraction. \end{matrix}$
  Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.
3. ⓒ
  $\begin{matrix} 6 - | 5 - 8 | + 3 (4 - 1) & = & 6 - | −3 | + 3 (3) & Simplify inside grouping symbols. \\ = & 6 - 3 + 3 (3) & Simplify absolute value. \\ = & 6 - 3 + 9 & Simplify multiplication. \\ = & 3 + 9 & Simplify subtraction. \\ = & 12 & Simplify addition. \end{matrix}$
4. ⓓ
  $\begin{matrix} \frac{14 - 3 \cdot 2}{2 \cdot 5 - 3^{2}} & = & \frac{14 - 3 \cdot 2}{2 \cdot 5 - 9} & Simplify exponent. \\ = & \frac{14 - 6}{10 - 9} & Simplify products. \\ = & \frac{8}{1} & Simplify differences. \\ = & 8 & Simplify quotient. \end{matrix}$
  In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.
5. ⓔ
  $\begin{matrix} 7 (5 \cdot 3) - 2 [(6 - 3) - 4^{2}] + 1 & = & 7 (15) - 2 [(3) - 4^{2}] + 1 & Simplify inside parentheses. \\ = & 7 (15) - 2 (3 - 16) + 1 & Simplify exponent. \\ = & 7 (15) - 2 (−13) + 1 & Subtract. \\ = & 105 + 26 + 1 & Multiply. \\ = & 132 & Add. \end{matrix}$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Use the order of operations to evaluate each of the following expressions.

ⓐ $\sqrt{5^{2} - 4^{2}} + 7 {(5 - 4)}^{2}$
ⓑ $1 + \frac{7 \cdot 5 - 8 \cdot 4}{9 - 6}$
ⓓ $\frac{1}{2} [5 \cdot 3^{2} - 7^{2}] + \frac{1}{3} \cdot 9^{2}$
ⓔ $[{(3 - 8)}^{2} - 4] - (3 - 8)$

How did it go?

Solution

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