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Identifying the base of an exponent incorrectly can cause the result to be incorrect. The error most often occurs with expressions of the form -32 and (-3)2. While these expressions look very similar, they represent two very different things. Expression Verbal Representation Mathematical Representation Base -32 Opposite of 3 squared -32 = -(3 ∙ 3) = -(9) or -9 3 (-3)2 Negative 3 squared (-3)2 = (-3)(-3) = 9 -3 The examples in the table illustrate the following facts. 1. If a negative sign is in front of an exponential expression, the negative sign indicates the opposite of the value of the exponential expression. In other words, the negative sign is not part of the base of the exponent. That is, -bn = -(b ∙ b ∙ b ∙ ∙ ∙ b) n times The base b is repeated as a factor n times. 2. If there are parentheses around a negative number raised to an exponent, then the exponent is applied to the negative number. The negative number is repeated as a factor. That is, (-b)n = (-b)(-b)(-b) ∙ ∙ ∙ (-b) n times The base -b is repeated as a factor n times. ���������������������� Simplifying an Exponential Expression Step 1: Identify the base and exponent. Step 2: Rewrite the expression as repeated multiplication. Step 3: Simplify the result. �� ������������������������ ������������������ Identify the base, determine if the negative sign repeats in the product, and simplify each exponential expression. Problems Base Does negative sign repeat? Evaluate 2a. (-4)3 -4 yes (-4)3 = (-4)(-4)(-4) = -64 2b. -54 5 no -54 = -(5 ∙ 5 ∙ 5 ∙ 5) = -625 2c. a- 1 2 b 2 - 1 2 yes a- 1 2 b 2 = a- 1 2 ba- 1 2 b = 1 4 2d. -(-3)5 -3 yes We must find the opposite of -3 raised to the fifth power. -(-3)5 = -(-3)(-3)(-3)(-3)(-3) = -(-243) = 243 60 Chapter 1 Real Numbers and Algebraic Expressions


hendricks_beginning_algebra_1e_ch1_3
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