Solve:
$\left(\frac{1}{8}\right)^{-3 a}=512^{3 a}$
A. $a=-8$
B. $a=0$
C. $a=8$
D. no solution
Answer (2)
Rewrite both sides of the equation with the same base: 2 9 a = 2 27 a .
Equate the exponents: 9 a = 27 a .
Solve for a : 18 a = 0 .
The solution is a = 0 , which satisfies the original equation: 0 .
Explanation
Rewrite with the same base We are given the equation ( 8 1 ) − 3 a = 51 2 3 a and asked to solve for a . We need to rewrite both sides of the equation with the same base. Since 8 = 2 3 and 512 = 2 9 , we can rewrite the equation as ( 2 − 3 ) − 3 a = ( 2 9 ) 3 a .
Simplify the exponents Now, we simplify the exponents. On the left side, we have ( 2 − 3 ) − 3 a = 2 ( − 3 ) ( − 3 a ) = 2 9 a . On the right side, we have ( 2 9 ) 3 a = 2 ( 9 ) ( 3 a ) = 2 27 a . So, the equation becomes 2 9 a = 2 27 a .
Equate the exponents Since the bases are equal, the exponents must be equal. Therefore, we have 9 a = 27 a .
Solve for a Now, we solve for a . Subtracting 9 a from both sides gives 27 a − 9 a = 0 , which simplifies to 18 a = 0 .
The solution Dividing both sides by 18, we get a = 18 0 = 0 .
Verification We check the solution a = 0 in the original equation: ( 8 1 ) − 3 ( 0 ) = 51 2 3 ( 0 ) . This simplifies to ( 8 1 ) 0 = 51 2 0 , which further simplifies to 1 = 1 . Thus, a = 0 is a valid solution.
Examples
Imagine you're adjusting the settings on an audio equalizer. The problem above is similar to finding the correct setting ( a ) that balances two different frequency amplifications. Understanding exponential equations helps in various scenarios, such as adjusting audio settings, calibrating instruments, or balancing resource allocation. This algebraic approach ensures the system is correctly tuned and balanced.
We solved the equation ( 8 1 ) − 3 a = 51 2 3 a by rewriting both sides with base 2, leading to the equation 2 9 a = 2 27 a . This allowed us to equate the exponents and solve for a , resulting in a = 0 . The correct answer is option B: a = 0 .
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