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$Solve the exponential equation for x. {:[7^(-9x+1)=49^(-12)],[x=◻]:}$

Solve the exponential equation for $x$ . $\newline$ $\begin{array}{l} 7^{-9 x+1}=49^{-12} \\ x=\square \end{array}$

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Solve multi-step inequalities

Full solution

Q. Solve the exponential equation for

x

.

\newline

\begin{array}{l} 7^{-9 x+1}=49^{-12} \\ x=\square \end{array}

Recognizing the power of $7$ : First, we need to recognize that $49$ is a power of $7$ , specifically $49$ is $7$ squared ( $7^2$ ). This will allow us to rewrite the equation with the same base on both sides. $\newline$ $7^{(-9x+1)} = 49^{(-12)}$ $\newline$ $7^{(-9x+1)} = (7^2)^{(-12)}$
Applying the power of a power rule: Next, we apply the power of a power rule, which states that $(a^b)^c = a^{(b*c)}$ . We will use this to simplify the right side of the equation. $\newline$ $7^{(-9x+1)} = 7^{(2*(-12))}$ $\newline$ $7^{(-9x+1)} = 7^{-24}$
Setting the exponents equal: Since the bases are the same and the equation is an equality, we can set the exponents equal to each other. $\newline$ $-9x + 1 = -24$
Isolating the variable: Now, we will solve for $x$ by isolating the variable. First, we subtract $1$ from both sides of the equation. $-9x + 1 - 1 = -24 - 1$ $-9x = -25$
Solving for x: Finally, we divide both sides by $-9$ to solve for $x$ .
$-9x / -9 = -25 / -9$
$x = 25/9$

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