College Algebra (10th Edition)

Published by Pearson
ISBN 10: 0321979478
ISBN 13: 978-0-32197-947-6

Chapter 6 - Section 6.3 - Exponential Functions - 6.3 Assess Your Understanding: 76

Answer

The solution set is $\left\{2, 4\right\}$.

Work Step by Step

To solve the given equation, make the two sides have the same base. Note that $125=5^3$, so the given equation is equivalent to: $5^{x^2+8}=(5^3)^{2x}$ Use the rule $(a^m)^n = a^{mn}$ to obtain: $5^{x^2+8} = 5^{3(2x)} \\5^{x^2+8}=5^{6x}$ Use the rule $a^m=a^n \longrightarrow m=n$ to obtain: $x^2+8=6x$ Subtract $6x$ to both sides of the equation to obtain: $\begin{array}{ccc} &x^2+8-6x &= &6x-6x \\&x^2-6x+8 &= &0 \end{array}$ Factor the trinomial to obtain: $(x-4)(x-2)=0$ Equate each factor to zero, and then solve each equation to obtain: $\begin{array}{ccc} &x-4=0 &\text{ or } &x-2-0 \\&x=4 &\text{ or } &x=2 \end{array}$ Thus, the solution set is $\left\{2, 4\right\}$.
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