To solve the equation [tex]\( 5^{-4x} = 25^{1-x} \)[/tex], we'll start by rewriting everything with a common base. Notice that [tex]\(25\)[/tex] can be expressed as a power of [tex]\(5\)[/tex].
Since [tex]\(25 = 5^2\)[/tex], we can rewrite the equation as:
[tex]\[
5^{-4x} = (5^2)^{1-x}
\][/tex]
Next, apply the power of a power property, [tex]\((a^b)^c = a^{bc}\)[/tex], to the right-hand side:
[tex]\[
(5^2)^{1-x} = 5^{2(1-x)}
\][/tex]
This simplifies to:
[tex]\[
5^{-4x} = 5^{2 - 2x}
\][/tex]
Now that both sides of the equation have the same base, we can set their exponents equal to each other:
[tex]\[
-4x = 2 - 2x
\][/tex]
To solve for [tex]\( x \)[/tex], first isolate the terms involving [tex]\( x \)[/tex] on one side of the equation. Add [tex]\( 2x \)[/tex] to both sides:
[tex]\[
-4x + 2x = 2 - 2x + 2x
\][/tex]
Simplify:
[tex]\[
-2x = 2
\][/tex]
Now, divide both sides by [tex]\(-2\)[/tex]:
[tex]\[
x = \frac{2}{-2}
\][/tex]
This simplifies to:
[tex]\[
x = -1
\][/tex]
Therefore, the solution to the equation [tex]\( 5^{-4x} = 25^{1-x} \)[/tex] is:
[tex]\[
x = -1
\][/tex]
