Certainly! Let's solve the equation [tex]\(\left(\frac{1}{2}\right)^{2x} = 32\)[/tex] step-by-step to determine which of the given options represents the correct equation to find [tex]\( x \)[/tex].
1. Start with the given equation:
[tex]\[
\left(\frac{1}{2}\right)^{2x} = 32
\][/tex]
2. Rewrite 32 as a power of 2:
[tex]\[
32 = 2^5
\][/tex]
Thus, the equation becomes:
[tex]\[
\left(\frac{1}{2}\right)^{2x} = 2^5
\][/tex]
3. Rewrite [tex]\(\left(\frac{1}{2}\right)\)[/tex] as [tex]\(2^{-1}\)[/tex]:
[tex]\[
\left(2^{-1}\right)^{2x} = 2^5
\][/tex]
Using the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we get:
[tex]\[
2^{-2x} = 2^5
\][/tex]
4. Since the bases are the same, set the exponents equal to each other:
[tex]\[
-2x = 5
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{5}{-2}
\][/tex]
Simplifying the fraction:
[tex]\[
x = -\frac{5}{2}
\][/tex]
6. Finally, interpret the correct equation form:
[tex]\[
-2x = 5
\][/tex]
Thus, the correct equation to find the solution of [tex]\(\left(\frac{1}{2}\right)^{2 \cdot x} = 32\)[/tex] is:
[tex]\[
-2x = 5
\][/tex]
The correct option is:
[tex]\[
-2x = 5
\][/tex]
Hence, the answer is:
[tex]\[
-2 x = 5
\][/tex]
