Section 1.3 The Order of Operations, Algebraic Expressions, and Equations 27 105. Multiply and simplify the result: 2 �� ���������������� �������� The Order of Operations, Algebraic Expressions, and Equations While we may not simplify numerical expressions on a daily basis, many of the things we encounter in life are based on this skill. One example involves saving money. There is an equation that can be used to calculate how much money we will accumulate if we invest a specific amount of money in a savings account. For example, if we invest $5000 in a savings account that earns 6% annual interest for 3 yr, the total amount saved is given by the equation Amount saved = 5000(1.06)3 To find the value of this expression, we must know the order in which to perform the operations. In this section, we will learn how to simplify numerical expressions using the accepted order of operations. We will also learn how to apply this skill to evaluate algebraic expressions and equations. Exponential Expressions The equation shown in the section opener contains an expression of the form (1.06)3. This is an example of an exponential expression. (1.06)3 (1.06) is called the base. 3 is called an exponent. The exponent implies that we use the base as a repeated factor 3 times. That is, (1.06)3 = (1.06)(1.06)(1.06) ≈ 1.19 ������������������������ Exponential Notation For b a real number and n a natural number, bn = b · b · b . . . b ('')''* n times The number b is called the base of the exponential expression and the number n is called the exponent. ▶ OBJECTIVES As a result of completing this section, you will be able to 1. Evaluate exponential expressions. 2. Use the order of operations to simplify numerical expressions. 3. Evaluate algebraic expressions. 4. Determine if a value is a solution of an equation. 5. Express relationships mathematically. 6. Solve application problems. 7. Troubleshoot common errors. Objective 1 ▶ Evaluate exponential expressions. 1 4 3 6 3 5 . Scott’s work: 2 1 3 3 3 3 4 = 6 3 12 = 6 1 4 = 25 4 106. Subtract and simplify the result: 3 8 - 1 12 . Pauline’s work: 8 3 12 = 96 3 8 - 1 12 = 3 3 12 8 3 12 - 1 3 8 12 3 8 = 36 96 - 8 96 = 28 96 = 4 3 7 4 3 24 = 7 24 Calculate It! Use a calculator to perform the operation. Verify the answer by performing the operation by hand. 107. 3 7 + 12 11 108. 8 13 · 6 109. 3 - 4 3 110. 12 35 · 14 3 111. 3 14 · 3 112. 5 + 6 5
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