What is the value of $A$ when we rewrite $\left(\frac{32}{5}\right)^{-2 x}$ as $A^{x}$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $A=-\frac{64}{5}$ $\newline$ B $A=\frac{1}{2}$ $\newline$ (C) $A=\left(\frac{5}{32}\right)^{-2}$ $\newline$ (D) $A=\left(\frac{32}{5}\right)^{-2}$

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Algebra 1

Compare linear and exponential growth

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Q. What is the value of

A

when we rewrite

\left(\frac{32}{5}\right)^{-2 x}

A^{x}

\newline

Choose

1

answer:

\newline

(A)

A=-\frac{64}{5}

\newline

A=\frac{1}{2}

\newline

(C)

A=\left(\frac{5}{32}\right)^{-2}

\newline

(D)

A=\left(\frac{32}{5}\right)^{-2}

Identify Base A: We need to express $\left(\frac{32}{5}\right)^{-2x}$ in the form of $A^{x}$ . To do this, we will find a base $A$ such that $A^{x}$ is equivalent to $\left(\frac{32}{5}\right)^{-2x}$ .
Focus on $(\frac{32}{5}):</b> First, let's focus on the base of the exponent, which is \$(\frac{32}{5})$ . We want to find $A$ such that $A^{x} = (\frac{32}{5})^{-2x}$ . To do this, we need to find a base that, when raised to the power of $x$ , gives the same result as $(\frac{32}{5})$ raised to the power of $-2x$ .
Rewrite as $(32/5)^{-2}$ : We can rewrite $\left(\frac{32}{5}\right)^{-2x}$ as $\left(\frac{32}{5}\right)^{-2}$ ^x by using the property of exponents that states $(b^{m})^{n} = b^{mn}$ . This means we are looking for $A$ such that $A = \left(\frac{32}{5}\right)^{-2}$ .
Calculate $(32/5)^{-2}$ : Now, let's calculate $((32)/(5))^{-2}$ . This is the same as $(5/32)^2$ because a negative exponent indicates reciprocal, and then we square the result.
Compare with Answer Choices: Calculating $(\frac{5}{32})^2$ gives us $(5^2)/(32^2) = \frac{25}{1024}$ . This is the value of $A$ .
Compare with Answer Choices: Calculating $(5/32)^2$ gives us $(5^2)/(32^2) = 25/1024$ . This is the value of $A$ .Now we compare the value of $A$ we found with the answer choices. The value of $A$ is $25/1024$ , which is not listed in the answer choices directly. However, we can see that this is equivalent to $((5)/(32))^2$ , which is the reciprocal of the base raised to the power of $-2$ . Therefore, the correct answer choice is $(C) A=((5)/(32))^(-2)$ .

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