To solve the equation [tex]\(25^{n+3} = \frac{1}{0.04}\)[/tex], we can follow a sequence of logical steps:
1. Simplify the Right-Hand Side:
Let's start by simplifying the right-hand side of the equation [tex]\(\frac{1}{0.04}\)[/tex].
[tex]\[
\frac{1}{0.04} = \frac{1}{\frac{4}{100}} = \frac{1}{\frac{1}{25}} = 25
\][/tex]
Therefore, the equation becomes:
[tex]\[
25^{n+3} = 25
\][/tex]
2. Rewrite the Right-Hand Side as an Exponential Expression:
We know [tex]\(25\)[/tex] can be written as an exponential term with base 25:
[tex]\[
25 = 25^1
\][/tex]
Now the equation is:
[tex]\[
25^{n+3} = 25^1
\][/tex]
3. Set the Exponents Equal to Each Other:
Since the bases on both sides of the equation are the same, we can set the exponents equal to each other:
[tex]\[
n + 3 = 1
\][/tex]
4. Solve for [tex]\(n\)[/tex]:
To find the value of [tex]\(n\)[/tex], solve the equation [tex]\(n + 3 = 1\)[/tex]:
[tex]\[
n + 3 = 1
\][/tex]
Subtract 3 from both sides of the equation:
[tex]\[
n = 1 - 3
\][/tex]
[tex]\[
n = -2
\][/tex]
Therefore, the solution for the equation [tex]\(25^{n+3} = \frac{1}{0.04}\)[/tex] is [tex]\(n = -2\)[/tex].
