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$Solve the exponential equation for x. {:[3^(7-2x)=((1)/(27))^(-8)],[x=]:}$

Solve the exponential equation for $x$ . $\newline$ $\begin{array}{l} 3^{7-2 x}=\left(\frac{1}{27}\right)^{-8} \\ x=\square \end{array}$

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Solve multi-step inequalities

Full solution

Q. Solve the exponential equation for

x

.

\newline

\begin{array}{l} 3^{7-2 x}=\left(\frac{1}{27}\right)^{-8} \\ x=\square \end{array}

Recognize Exponential Form: First, we need to recognize that both sides of the equation are written in exponential form, and we can use the property of exponents to simplify the equation. The equation is $3^{7-2x} = (\frac{1}{27})^{-8}$ .
Rewrite Using Property: We know that $27$ is $3^3$ , so we can rewrite $\frac{1}{27}$ as $3^{-3}$ . Therefore, $(\frac{1}{27})^{-8}$ becomes $(3^{-3})^{-8}$ .
Simplify Right Side: Using the power of a power property $(a^{m})^{n} = a^{m*n}$ , we can simplify the right side of the equation: $(3^{-3})^{-8} = 3^{-3*-8} = 3^{24}$ .
Set Exponents Equal: Now the equation is $3^{7-2x} = 3^{24}$ . Since the bases are the same, we can set the exponents equal to each other: $7 - 2x = 24$ .
Isolate and Subtract: To solve for $x$ , we need to isolate $x$ . We can start by subtracting $7$ from both sides of the equation: $7 - 2x - 7 = 24 - 7$ .
Divide to Solve: This simplifies to $-2x = 17$ . Now we divide both sides by $-2$ to solve for $x$ : $\frac{-2x}{-2} = \frac{17}{-2}$ .
Final Solution: After dividing, we get $x = -\frac{17}{2}$ or $x = -8.5$ .

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