solve: $49^{3x}=343^{2x + 1}$\n$x=-3$\n$x = 1$\n$x = 3$\nno solution

Explanation:

Step1: Rewrite bases as powers of 7

Since $49 = 7^2$ and $343=7^3$, the equation $49^{3x}=343^{2x + 1}$ can be rewritten as $(7^2)^{3x}=(7^3)^{2x + 1}$.

Step2: Apply power - of - a - power rule

Using the rule $(a^m)^n=a^{mn}$, we get $7^{2\times3x}=7^{3\times(2x + 1)}$, which simplifies to $7^{6x}=7^{6x+3}$.

Step3: Set exponents equal

If $a^m=a^n$, then $m = n$. So, $6x=6x + 3$.

Step4: Solve for x

Subtract $6x$ from both sides: $6x-6x=6x + 3-6x$, resulting in $0 = 3$, which is a contradiction.

Answer:

no solution