Answer :
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]
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]
