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$Solve the exponential equation for x. {:[32^((x)/(5))=((1)/(16))^(4x-3)],[x=◻]:}$

Solve the exponential equation for $x$ . $\newline$ $\begin{array}{l} 32^{\frac{x}{5}}=\left(\frac{1}{16}\right)^{4 x-3} \\ x=\square \end{array}$

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Evaluate exponential functions

Full solution

Q. Solve the exponential equation for

x

.

\newline

\begin{array}{l} 32^{\frac{x}{5}}=\left(\frac{1}{16}\right)^{4 x-3} \\ x=\square \end{array}

Write Given Exponential Equation: Write down the given exponential equation. $\newline$ The equation is $32^{\frac{x}{5}} = \left(\frac{1}{16}\right)^{4x-3}$ .
Recognize Powers of $2$ : Recognize that $32$ and $1$ / $16$ can be written as powers of $2$ . $\newline$ $32 = 2^5$ and $1/16 = 2^{-4}$ . $\newline$ Rewrite the equation using these powers of $2$ . $\newline$ $2^{5 \cdot \frac{x}{5}} = 2^{-4 \cdot (4x-3)}$ .
Rewrite Using Powers of $2$ : Simplify the exponents. $\newline$ Since $5 \cdot \frac{x}{5} = x$ and $-4 \cdot (4x-3) = -16x + 12$ , the equation becomes: $\newline$ $2^x = 2^{-16x + 12}$ .
Simplify Exponents: Since the bases are the same, set the exponents equal to each other. $\newline$ $x = -16x + 12$ .
Set Exponents Equal: Solve for x. $\newline$ Add $16$ x to both sides of the equation to get all x terms on one side: $\newline$ $x + 16x = 12$ . $\newline$ Combine like terms: $\newline$ $17x = 12$ .
Solve for x: Divide both sides by $17$ to isolate x. $\newline$ $x = \frac{12}{17}$ .

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