For what value of $a$ does $\left(\frac{1}{9}\right)^{a+1}=81^{a+1} \cdot 27^{2-a} ?$
A. -4
B. -2
C. 2
D. 6
Answer (1)
Rewrite the equation using the base 3.
Simplify the exponents.
Combine the exponents on the right side.
Equate the exponents and solve for a : − 4 .
Explanation
Analyze the problem We are given the equation ( 9 1 ) a + 1 = 8 1 a + 1 ⋅ 2 7 2 − a . We need to find the value of a that satisfies this equation. Notice that 9 1 , 81 , and 27 are all powers of 3 . Specifically, 9 1 = 3 − 2 , 81 = 3 4 , and 27 = 3 3 .
Rewrite in terms of base 3 We can rewrite the given equation in terms of base 3: ( 3 − 2 ) a + 1 = ( 3 4 ) a + 1 ⋅ ( 3 3 ) 2 − a . Using the power of a power rule, we have 3 − 2 ( a + 1 ) = 3 4 ( a + 1 ) ⋅ 3 3 ( 2 − a ) .
Combine exponents Using the property x m ⋅ x n = x m + n , we can combine the exponents on the right side: 3 − 2 ( a + 1 ) = 3 4 ( a + 1 ) + 3 ( 2 − a ) .
Equate the exponents Since the bases are equal, we can equate the exponents: − 2 ( a + 1 ) = 4 ( a + 1 ) + 3 ( 2 − a ) . Expanding the terms, we get − 2 a − 2 = 4 a + 4 + 6 − 3 a .
Solve for a Simplifying the equation, we have − 2 a − 2 = a + 10. Adding 2 a to both sides gives − 2 = 3 a + 10. Subtracting 10 from both sides gives − 12 = 3 a . Dividing both sides by 3, we find a = − 4.
Verify the answer To verify our answer, we substitute a = − 4 into the original equation: ( 9 1 ) − 4 + 1 = 8 1 − 4 + 1 ⋅ 2 7 2 − ( − 4 ) ( 9 1 ) − 3 = 8 1 − 3 ⋅ 2 7 6 ( 3 − 2 ) − 3 = ( 3 4 ) − 3 ⋅ ( 3 3 ) 6 3 6 = 3 − 12 ⋅ 3 18 3 6 = 3 − 12 + 18 3 6 = 3 6 Since the equation holds true, our answer is correct.
Final Answer Therefore, the value of a that satisfies the equation is a = − 4 .
Examples
Exponential equations are used in various fields such as finance, physics, and computer science. For example, in finance, they are used to model compound interest. Suppose you invest P dollars in an account that pays an annual interest rate r compounded n times per year. The amount A you will have after t years is given by the formula A = P ( 1 + n r ) n t . Solving for t or r in such equations often involves using logarithms, which are closely related to exponential functions. Understanding how to manipulate and solve exponential equations is crucial for making informed financial decisions.
