Alright, let's solve the equation [tex]\(32 = 16^{-x+1}\)[/tex] for [tex]\( x \)[/tex].
### Step-by-Step Solution:
1. Write the equation:
[tex]\[
32 = 16^{-x + 1}
\][/tex]
2. Express both sides with the same base:
We know that [tex]\(32\)[/tex] and [tex]\(16\)[/tex] can be written as powers of [tex]\(2\)[/tex]:
[tex]\[
32 = 2^5
\][/tex]
[tex]\[
16 = 2^4
\][/tex]
So, substitute these into the equation:
[tex]\[
2^5 = (2^4)^{-x + 1}
\][/tex]
3. Simplify the right-hand side:
Apply the exponent rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(2^4)^{-x + 1} = 2^{4(-x + 1)}
\][/tex]
Which simplifies to:
[tex]\[
2^5 = 2^{4(-x + 1)}
\][/tex]
4. Equate the exponents:
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
5 = 4(-x + 1)
\][/tex]
5. Solve the equation for [tex]\( x \)[/tex]:
Distribute the [tex]\(4\)[/tex] on the right-hand side:
[tex]\[
5 = 4(-x) + 4
\][/tex]
[tex]\[
5 = -4x + 4
\][/tex]
6. Isolate [tex]\( x \)[/tex]:
Subtract [tex]\(4\)[/tex] from both sides:
[tex]\[
5 - 4 = -4x
\][/tex]
[tex]\[
1 = -4x
\][/tex]
Divide both sides by [tex]\(-4\)[/tex]:
[tex]\[
x = -\frac{1}{4}
\][/tex]
So, the solution to the equation [tex]\(32 = 16^{-x + 1}\)[/tex] is:
[tex]\[
x = -0.25
\][/tex]
