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

We rewrote the equation 4 9 3 x = 34 3 2 x + 1 using the common base of 7, yielding 7 6 x = 7 6 x + 3 . This led to a contradiction when equating the exponents, showing there are no solutions for x . Therefore, the original equation was simplified correctly using base 7 expressions.
;

Answered by Anonymous | 2025-07-04

Express 49 as 7 2 and 343 as 7 3 .
Rewrite the equation as ( 7 2 ) 3 x = ( 7 3 ) 2 x + 1 .
Simplify the exponents: 7 6 x = 7 6 x + 3 .
Both sides of the equation are written in terms of base 7: 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 both sides in terms of the same base. We recognize that both 49 and 343 are powers of 7.

Expressing in terms of base 7 We can express 49 as 7 2 and 343 as 7 3 . Substituting these into the original equation, we get ( 7 2 ) 3 x = ( 7 3 ) 2 x + 1 .

Simplifying the exponents Using the power of a power rule, which states that ( a m ) n = a mn , we simplify the exponents: 7 2 ( 3 x ) = 7 3 ( 2 x + 1 ) , which simplifies to 7 6 x = 7 6 x + 3 .

Equating the exponents Since the bases are now the same, we can equate the exponents: 6 x = 6 x + 3 .

Solving for x and Rewriting Subtracting 6 x from both sides, we get 0 = 3 , which is a contradiction. This means there is no solution for x that satisfies the original equation. However, the question asks to write each side of the equation in terms of base 7. Therefore, we have 4 9 3 x = ( 7 2 ) 3 x = 7 6 x and 34 3 2 x + 1 = ( 7 3 ) 2 x + 1 = 7 6 x + 3 .

Final Answer Thus, we have rewritten both sides of the equation in terms of base 7: 7 6 x = 7 6 x + 3 .


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. Understanding how to solve exponential equations allows us to predict future values and make informed decisions in these areas. For example, if we know the initial population of a bacteria colony and its growth rate, we can use an exponential equation to predict the population size at a later time.

Answered by GinnyAnswer | 2025-07-04