Question

resolver: $49^{3x} = 343^{2x + 1}$
$x = 1$
$x = 3$
$x = -3$
no hay solución

Explanation:

Step1: Express bases as powers of 7

49 is \(7^2\) and 343 is \(7^3\). So rewrite the equation: \((7^2)^{3x} = (7^3)^{2x + 1}\)

Step2: Apply exponent power rule (\((a^m)^n=a^{mn}\))

Simplify both sides: \(7^{6x} = 7^{6x + 3}\)

Step3: Set exponents equal (since bases are same and positive)

\(6x = 6x + 3\)

Step4: Solve for x

Subtract \(6x\) from both sides: \(0 = 3\), which is false. Wait, no—wait, did I make a mistake? Wait, no, wait: Wait, 49 is \(7^2\), so \((7^2)^{3x}=7^{6x}\). 343 is \(7^3\), so \((7^3)^{2x + 1}=7^{3(2x + 1)} = 7^{6x + 3}\). Then \(7^{6x}=7^{6x + 3}\) implies \(6x = 6x + 3\), so \(0 = 3\), which is impossible. But wait, the options have "No hay solución"? Wait, but let's check again. Wait, maybe I messed up the bases. Wait, 49 is \(7^2\), 343 is \(7^3\). So the original equation is \(7^{6x}=7^{6x + 3}\). Subtract \(7^{6x}\) from both sides: \(0 = 7^{6x}(7^3 - 1)\)? No, wait, no—when you have \(a^m = a^n\), then \(m = n\) if \(a>0, a
eq1\). So here, \(7>0, 7
eq1\), so \(6x = 6x + 3\) → \(0 = 3\), which is a contradiction. So there's no solution. Wait, but the options include "No hay solución". But wait, maybe I made a mistake in the exponent. Wait, 49 is \(7^2\), so \(49^{3x}=(7^2)^{3x}=7^{6x}\). 343 is \(7^3\), so \(343^{2x + 1}=(7^3)^{2x + 1}=7^{3(2x + 1)}=7^{6x + 3}\). So \(7^{6x}=7^{6x + 3}\) → \(6x = 6x + 3\) → \(0 = 3\), which is false. So the equation has no solution. But wait, the options: the last option is "No hay solución". But wait, maybe I made a mistake. Wait, let's check with x=-3: Left side: \(49^{3(-3)}=49^{-9}=\frac{1}{49^9}\). Right side: \(343^{2(-3)+1}=343^{-5}=\frac{1}{343^5}\). Are these equal? \(49^9=(7^2)^9=7^{18}\), \(343^5=(7^3)^5=7^{15}\). So \(\frac{1}{7^{18}}\) vs \(\frac{1}{7^{15}}\): not equal. x=3: Left: \(49^{9}=7^{18}\), Right: \(343^{7}=7^{21}\). Not equal. x=1: Left: \(49^3=7^6\), Right: \(343^3=7^9\). Not equal. So indeed, no solution. Wait, but the options have "No hay solución" as the last option. So the correct answer is "No hay solución". Wait, but let's confirm again. The equation \(49^{3x}=343^{2x + 1}\) is \(7^{6x}=7^{6x + 3}\), which implies \(6x = 6x + 3\), so 0=3, which is impossible. Thus, no solution.

Answer:

No hay solución