To solve the equation [tex]\(\left(\frac{1}{4}\right)^{x+1}=32\)[/tex], we need to find the value of [tex]\(x\)[/tex].
We can start by expressing both sides of the equation using the same base:
[tex]\[
\left(\frac{1}{4}\right)^{x+1} = 32
\][/tex]
First, note that [tex]\(\frac{1}{4}\)[/tex] can be written as [tex]\(4^{-1}\)[/tex] and [tex]\(32\)[/tex] can be written as [tex]\(2^5\)[/tex]. Hence, we rewrite the equation as:
[tex]\[
(4^{-1})^{x+1} = 2^5
\][/tex]
Next, we need to express [tex]\(4\)[/tex] in terms of base [tex]\(2\)[/tex]. Since [tex]\(4 = 2^2\)[/tex], we have:
[tex]\[
(2^{-2})^{x+1} = 2^5
\][/tex]
Simplify the left-hand side:
[tex]\[
2^{-2(x+1)} = 2^5
\][/tex]
Now that the bases are the same, we can equate the exponents:
[tex]\[
-2(x+1) = 5
\][/tex]
Distribute the [tex]\(-2\)[/tex] on the left-hand side:
[tex]\[
-2x - 2 = 5
\][/tex]
Solve for [tex]\(x\)[/tex] by isolating it on one side:
[tex]\[
-2x - 2 = 5
\][/tex]
[tex]\[
-2x = 5 + 2
\][/tex]
[tex]\[
-2x = 7
\][/tex]
[tex]\[
x = -\frac{7}{2}
\][/tex]
Thus, the correct solution to the equation is:
[tex]\[
\boxed{-\frac{7}{2}}
\][/tex]
So the correct answer is option B: [tex]\(-\frac{7}{2}\)[/tex].
