To solve $49^{3x} = 343^{2x+1}$, write each side of the equation in terms of base ____.

To rewrite the equation 4 9 3 x = 34 3 2 x + 1 in terms of base 7, we find that 49 = 7 2 and 343 = 7 3 . This changes the equation to 7 6 x = 7 6 x + 3 , simplifying the problem. From here, we can solve for the variable x directly.
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Answered by Anonymous | 2025-07-04

Rewrite 49 and 343 as powers of 7 : 49 = 7 2 and 343 = 7 3 .
Substitute these into the original equation: ( 7 2 ) 3 x = ( 7 3 ) 2 x + 1 .
Simplify the exponents using the power of a power rule: 7 6 x = 7 6 x + 3 .
The equation with base 7 is 7 6 x = 7 6 x + 3 .

Explanation

Problem Analysis We are given the equation 4 9 3 x = 34 3 2 x + 1 and asked to rewrite each side in terms of base 7.

Rewriting with Base 7 We know that 49 = 7 2 and 343 = 7 3 . Substituting these into the given equation, we have ( 7 2 ) 3 x = ( 7 3 ) 2 x + 1 .

Simplifying Exponents Using the power of a power rule, we simplify the exponents: 7 2 ⋅ 3 x = 7 3 ⋅ ( 2 x + 1 ) , which gives 7 6 x = 7 6 x + 3 .

Final Answer Therefore, the left side of the equation in terms of base 7 is 7 6 x and the right side is 7 6 x + 3 .


Examples
Exponential equations are used in various fields such as finance, physics, and computer science. For example, in finance, compound interest is calculated using exponential functions. If you invest P dollars at an annual interest rate r compounded n times per year, the amount A you will have after t years is given by A = P ( 1 + n r ​ ) n t . Understanding exponential equations helps in predicting the growth of investments over time.

Answered by GinnyAnswer | 2025-07-04