1.1 Real numbers: algebra essentials (Page 3/35)

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Using the order of operations

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

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

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

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Try It

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

$\sqrt{5^{2} - 4^{2}} + 7 {(5 - 4)}^{2}$
$1 + \frac{7 \cdot 5 - 8 \cdot 4}{9 - 6}$
$| 1.8 - 4.3 | + 0.4 \sqrt{15 + 10}$
$\frac{1}{2} [5 \cdot 3^{2} - 7^{2}] + \frac{1}{3} \cdot 9^{2}$
$[{(3 - 8)}^{2} - 4] - (3 - 8)$

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Using properties of real numbers

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.

Commutative properties

The commutative property of addition states that numbers may be added in any order without affecting the sum.

a + b = b + a

We can better see this relationship when using real numbers.

\begin{array}{l} (−2) + 7 = 5 & and & 7 + (−2) = 5 \end{array}

Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.

a \cdot b = b \cdot a

Again, consider an example with real numbers.

\begin{matrix} (−11) \cdot (−4) = 44 & and & (−4) \cdot (−11) = 44 \end{matrix}

It is important to note that neither subtraction nor division is commutative. For example, $17 - 5$ is not the same as $5 - 17.$ Similarly, $20 \div 5 \neq 5 \div 20.$

Associative properties

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.

a (b c) = (a b) c

Consider this example.

\begin{matrix} (3 \cdot 4) \cdot 5 = 60 & and & 3 \cdot (4 \cdot 5) = 60 \end{matrix}

The associative property of addition tells us that numbers may be grouped differently without affecting the sum.

a + (b + c) = (a + b) + c

This property can be especially helpful when dealing with negative integers. Consider this example.

\begin{matrix} [15 + (−9)] + 23 = 29 & and & 15 + [(−9) + 23] = 29 \end{matrix}

Are subtraction and division associative? Review these examples.

\begin{matrix} 8 - (3 - 15) & \overset{?}{=} & (8 - 3) - 15 & 64 \div (8 \div 4) & \overset{?}{=} & (64 \div 8) \div 4 \\ 8 - (- 12) & = & 5 - 15 & 64 \div 2 & \overset{?}{=} & 8 \div 4 \\ 20 & \neq & 20 - 10 & 32 & \neq & 2 \end{matrix}

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Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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