If [tex]\frac{1}{64}=4^{2 s-1} \cdot 16^{2 s+2}[/tex], what is the value of [tex]s[/tex]?
A. -1
B. 0
C. 1
D. no solution
Answer (1)
Rewrite the equation using powers of 4: 64 1 = 4 − 3 and 16 = 4 2 .
Simplify the equation: 4 − 3 = 4 2 s − 1 ⋅ 4 4 s + 4 .
Combine the exponents: 4 − 3 = 4 6 s + 3 .
Equate the exponents and solve for s : − 3 = 6 s + 3 , which gives s = − 1 .
− 1
Explanation
Problem Analysis We are given the equation 64 1 = 4 2 s − 1 ⋅ 1 6 2 s + 2 and we need to find the value of s .
Rewriting with Powers of 4 First, we can rewrite the equation using powers of 4. Since 64 = 4 3 , we have 64 1 = 4 3 1 = 4 − 3 . Also, 16 = 4 2 , so we can rewrite the equation as: 4 − 3 = 4 2 s − 1 ⋅ ( 4 2 ) 2 s + 2
Simplifying the Equation Next, we simplify the equation using the exponent rule ( a m ) n = a mn : 4 − 3 = 4 2 s − 1 ⋅ 4 2 ( 2 s + 2 ) = 4 2 s − 1 ⋅ 4 4 s + 4
Combining Exponents Now, we use the exponent rule a m ⋅ a n = a m + n to combine the terms on the right side: 4 − 3 = 4 2 s − 1 + 4 s + 4 = 4 6 s + 3
Equating Exponents Since the bases are equal, we can equate the exponents: − 3 = 6 s + 3
Solving for s Now, we solve for s : 6 s + 3 = − 3 \6 s = − 3 − 3 \6 s = − 6 \s = 6 − 6 \s = − 1
Final Answer Therefore, the value of s is − 1 .
Examples
Understanding exponential equations is crucial in many fields, such as finance and computer science. For example, calculating the depreciation of an asset or the growth of an investment often involves exponential functions. If a car's value depreciates at a rate described by an exponential equation, solving for the time it takes to reach a certain value uses similar algebraic manipulations as in this problem. This skill is also fundamental in understanding algorithms' time complexity in computer science, where exponential growth can significantly impact performance.
