Which equation is equivalent to $16^{2 p}=32^{p+3}$?
A. $8^{4 p}=8^{4 p+3}$
B. $8^{4 p}=8^{4 p+12}$
C. $2^{8 p}=2^{5 p+15}$
D. $2^{8 p}=2^{5 p+3}$
Answer (1)
Express both sides of the equation as powers of 2.
Simplify the equation using the power of a power rule: 2 8 p = 2 5 p + 15 .
Compare the simplified equation with the given options.
Identify the matching equation: 2 8 p = 2 5 p + 15 .
Explanation
Understanding the Problem We are given the equation 1 6 2 p = 3 2 p + 3 and asked to find an equivalent equation from the given options.
Expressing as Powers of 2 We can express both sides of the equation as powers of 2. Recall that 16 = 2 4 and 32 = 2 5 .
Simplifying the Equation Substituting these into the original equation, we have ( 2 4 ) 2 p = ( 2 5 ) p + 3 . Using the power of a power rule, we get 2 4 ( 2 p ) = 2 5 ( p + 3 ) , which simplifies to 2 8 p = 2 5 p + 15 .
Comparing with Options Now we compare this equation, 2 8 p = 2 5 p + 15 , with the given options:
8 4 p = 8 4 p + 3
8 4 p = 8 4 p + 12
2 8 p = 2 5 p + 15
2 8 p = 2 5 p + 3
The third option, 2 8 p = 2 5 p + 15 , matches our simplified equation.
Final Answer Therefore, the equation equivalent to 1 6 2 p = 3 2 p + 3 is 2 8 p = 2 5 p + 15 .
Examples
Exponential equations are used in various fields such as finance, physics, and computer science. For example, in finance, compound interest calculations involve exponential growth. If you invest P dollars at an annual interest rate r compounded n times per year, the amount A you'll have after t years is given by A = P ( 1 + n r ) n t . Understanding how to manipulate exponential equations allows you to solve for any of these variables, such as determining how long it will take for your investment to double.
