Which equation is equivalent to $16^{2 k}=32^{k+3}$?
A. $8^{4 k}=8^{4 k+3}$
B. $8^{4 k}=8^{4 k+6}$
C. $2^{8 k}=2^{5 k+15}$
D. $2^{8 k}=2^{5 k+3}$
Answer (1)
Express both sides of the equation with the same base: 1 6 2 k = 3 2 k + 3 becomes ( 2 4 ) 2 k = ( 2 5 ) k + 3 .
Simplify the exponents: 2 8 k = 2 5 ( k + 3 ) .
Further simplify: 2 8 k = 2 5 k + 15 .
The equivalent equation is 2 8 p = 2 5 p + 15 .
Explanation
Understanding the Problem We are given the equation 1 6 2 k = 3 2 k + 3 and asked to find an equivalent equation from the given options.
Expressing with the Same Base First, we express both sides of the given equation with the same base. We know that 16 = 2 4 and 32 = 2 5 . Therefore, we can rewrite the equation as ( 2 4 ) 2 k = ( 2 5 ) k + 3 .
Simplifying the Exponents Next, we simplify the exponents. Using the power of a power rule, we have 2 4 ⋅ 2 k = 2 5 ⋅ ( k + 3 ) , which simplifies to 2 8 k = 2 5 ( k + 3 ) .
Further Simplification Further simplification yields 2 8 k = 2 5 k + 15 .
Comparing with the Options Now, we compare this result with the given options:
Option 1: 8 4 x = 8 4 x + 3 Option 2: 8 4 , 2 = 8 4 , 3 + 12 Option 3: 2 8 p = 2 5 p + 15 Option 4: 2 8 p = 2 5 p + 3
We can see that option 3, 2 8 p = 2 5 p + 15 , matches our simplified equation 2 8 k = 2 5 k + 15 , where p is used instead of k .
Final Answer Therefore, the equation equivalent to 1 6 2 k = 3 2 k + 3 is 2 8 p = 2 5 p + 15 .
Examples
Exponential equations are used in various fields, such as finance to calculate compound interest, in biology to model population growth, and in physics to describe radioactive decay. For example, if a population of bacteria doubles every hour, the population size can be modeled by an exponential equation. Similarly, the decay of a radioactive substance can be modeled using an exponential equation, allowing scientists to determine the age of ancient artifacts through carbon dating. Understanding how to manipulate and solve exponential equations is crucial for making predictions and understanding these real-world phenomena.
