To solve the equation [tex]\(\frac{1}{25} = 5^{x+4}\)[/tex], let's follow these steps:
1. Rewrite the Left-Hand Side:
We know that 25 can be expressed as [tex]\(5^2\)[/tex]. Therefore, [tex]\(\frac{1}{25}\)[/tex] can be rewritten as [tex]\(\frac{1}{5^2}\)[/tex].
2. Use the Property of Exponents:
[tex]\(\frac{1}{5^2}\)[/tex] can be expressed as [tex]\(5^{-2}\)[/tex], since [tex]\(\frac{1}{a^n} = a^{-n}\)[/tex].
Now, the original equation becomes:
[tex]\[
5^{-2} = 5^{x+4}
\][/tex]
3. Compare the Exponents:
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
-2 = x + 4
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
To isolate [tex]\(x\)[/tex], subtract 4 from both sides of the equation:
[tex]\[
-2 - 4 = x
\][/tex]
[tex]\[
x = -6
\][/tex]
Therefore, the solution to the equation [tex]\(\frac{1}{25} = 5^{x+4}\)[/tex] is [tex]\(x = -6\)[/tex].
The correct answer is:
[tex]\[
x = -6
\][/tex]
