If $\frac{1}{64}=4^{2 s-1} \cdot 16^{2 s+2}$, what is the value of $s$?
Answer (1)
Express all terms in the equation as powers of 4.
Simplify the equation using exponent rules.
Equate the exponents.
Solve for s , which gives s = − 1 .
The value of s is − 1 .
Explanation
Understanding the Problem 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 .
Expressing Terms as Powers of 4 First, we express all terms as powers of 4. We know that 64 1 = 4 3 1 = 4 − 3 and 16 = 4 2 . Substituting these into the equation, we get:
4 − 3 = 4 2 s − 1 ⋅ ( 4 2 ) 2 s + 2
Simplifying the Equation Next, we simplify the right side of the equation using the exponent rule ( a m ) n = a mn :
4 − 3 = 4 2 s − 1 ⋅ 4 2 ( 2 s + 2 )
4 − 3 = 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 − 3 = 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 = − 6
s = 6 − 6
s = − 1
Final Answer Therefore, the value of s is − 1 .
Examples
Understanding exponential equations is crucial in various fields, such as calculating compound interest or modeling population growth. For instance, if a population doubles every year, the growth can be modeled by an exponential equation. Similarly, in finance, compound interest calculations rely on exponential growth. By solving these equations, we can predict future values and make informed decisions.
