To solve $49^{3 x}=343^{2 x+1}$, write each side of the equation in terms of base $\square$. Choose the correct equation.
A. $(7)^{3 x}=\left(7^3\right)^{2 x+1}$
B. $\left(7^2\right)^{3 x}=\left(7^3\right)^{2 x+1}$
C. $\left(7^2\right)^{3 x}=\left(7^4\right)^{2 x+1}$
Answer (2)
The equation 4 9 3 x = 34 3 2 x + 1 can be correctly rewritten as ( 7 2 ) 3 x = ( 7 3 ) 2 x + 1 . Thus, the chosen option is A. This allows us to solve the equation with a common base, simplifying the process.
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Recognize that both 49 and 343 can be expressed as powers of 7.
Rewrite 49 as 7 2 and 343 as 7 3 .
Substitute these expressions into the original equation: ( 7 2 ) 3 x = ( 7 3 ) 2 x + 1 .
The correct equation is ( 7 2 ) 3 x = ( 7 3 ) 2 x + 1 .
Explanation
Understanding the Problem We are given the equation 4 9 3 x = 34 3 2 x + 1 and asked to rewrite it in terms of a common base.
Finding a Common Base We need to find a base that both 49 and 343 can be expressed as powers of. Since 49 = 7 2 and 343 = 7 3 , we can use 7 as the common base.
Rewriting with Base 7 Now, we rewrite the equation using the base 7. We have 4 9 3 x = ( 7 2 ) 3 x and 34 3 2 x + 1 = ( 7 3 ) 2 x + 1 .
Substituting into the Equation Substituting these expressions into the original equation, we get ( 7 2 ) 3 x = ( 7 3 ) 2 x + 1 .
Final Answer Therefore, the correct equation is ( 7 2 ) 3 x = ( 7 3 ) 2 x + 1 .
Examples
Exponential equations are used in various fields such as finance to calculate compound interest, in physics to model radioactive decay, and in biology to model population growth. For example, 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 solve exponential equations helps in predicting the future value of investments or the decay rate of radioactive substances.
