Solve: $\left(\frac{1}{4}\right)^{3 z-1}=16^{z+2} \cdot 64^{z-2}$
Answer (1)
Rewrite all terms as powers of 2: ( 4 1 ) 3 z − 1 = ( 2 − 2 ) 3 z − 1 , 1 6 z + 2 = ( 2 4 ) z + 2 , 6 4 z − 2 = ( 2 6 ) z − 2 .
Simplify the exponents: 2 − 6 z + 2 = 2 4 z + 8 ⋅ 2 6 z − 12 .
Combine the exponents: 2 − 6 z + 2 = 2 10 z − 4 .
Solve for z : z = 8 3 .
Explanation
Problem Analysis We are given the equation ( 4 1 ) 3 z − 1 = 1 6 z + 2 ⋅ 6 4 z − 2 and we want to solve for z .
Rewriting as Powers of 2 First, we rewrite all terms as powers of 2. We have 4 1 = 2 − 2 , 16 = 2 4 , and 64 = 2 6 . Substituting these into the equation, we get ( 2 − 2 ) 3 z − 1 = ( 2 4 ) z + 2 ⋅ ( 2 6 ) z − 2 .
Simplifying Exponents Next, we simplify the exponents. We have 2 − 2 ( 3 z − 1 ) = 2 4 ( z + 2 ) ⋅ 2 6 ( z − 2 ) , which simplifies to 2 − 6 z + 2 = 2 4 z + 8 ⋅ 2 6 z − 12 .
Combining Exponents Now, we combine the exponents on the right side of the equation: 2 − 6 z + 2 = 2 ( 4 z + 8 ) + ( 6 z − 12 ) , which simplifies to 2 − 6 z + 2 = 2 10 z − 4 .
Equating Exponents Since the bases are equal, we set the exponents equal to each other: − 6 z + 2 = 10 z − 4 .
Solving for z Finally, we solve for z . Adding 6 z to both sides gives 2 = 16 z − 4 . Adding 4 to both sides gives 6 = 16 z . Dividing both sides by 16 gives z = 16 6 = 8 3 .
Final Answer Therefore, the solution is z = 8 3 .
Examples
Imagine you're adjusting the settings on a sound equalizer. Each slider affects the sound output exponentially. Solving equations like this helps you determine the precise adjustments needed to achieve the desired sound balance, ensuring that different frequencies are amplified or attenuated correctly. This is crucial in audio engineering, where precise control over sound is essential for creating high-quality recordings and performances.
