22 Chapter 1 Real Numbers and Algebraic Expressions Order of Operations Without a standard order to perform operations in numerical expressions, we could possibly get different values for the same numerical expression. Consider the expression 4 + 5(6). There are two possible methods for simplifying this expression. Method 1 Method 2 4 + 5(6) = 4 + 30 Multiply. 4 + 5(6) = 9(6) Add. = 34 Add. = 54 Multiply. Both of these values cannot be correct. The first method is the correct method to simplify the expression. We must use a special order to simplify numerical expressions. The accepted order is called the order of operations. It is stated as follows. Procedure: Order of Operations When simplifying a numerical expression, perform the operations in the following order. Step 1: Simplify expressions inside grouping symbols first. Grouping symbols include parentheses, brackets, absolute value symbols, square root symbols, and fraction bars. Step 2: Simplify any exponential expressions. Step 3: Perform multiplication or division in order from left to right. (Since division is multiplying by the reciprocal, one operation doesn’t take precedence over the other.) Step 4: Perform addition or subtraction in order from left to right. (Since subtraction is adding the opposite, one operation doesn’t take precedence over the other.) Objective 4 Examples Use the order of operations to simplify each expression. 4a. -2 - 8 4 - (-1) 4b. 5(-4)2 + 2(-4) - 6 4c. 1(7 - 4)2 + (1 - 5)2 4d. 2 - 59 - 3(4 - 6) ÷ 3 · 8 4e. 2 - 3u5 - 9u2 -4 - 14 + 12 Solutions 4a. -2 - 8 4 - (-1) = -2 + (-8) 4 + 1 = -10 5 Rewrite the numerator and denominator as addition. Add. =-2 Simplify. 4b. 5(-4)2 + 2(-4) - 6 = 5(16) + 2(-4) - 6 Simplify the exponent. = 80 + (-8) - 6 Multiply from left to right. = 72 - 6 Add the fi rst two numbers. = 66 Subtract. 4c. 1(7 - 4)2 + (1 - 5)2 = 1(3)2 + (-4)2 Simplify within parentheses. = 19 + 16 Simplify the exponent. = 125 Add = 5 Apply the square root. Objective 4 ▶ Use the order of operations to simplify an expression.
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