Sure, let's simplify the given expression step by step:
Given expression: [tex]\(\frac{5^{n+2} - 5^n}{5^{n+1} + 5^n}\)[/tex]
1. Rewrite the exponents using the properties of exponents:
- Recall that [tex]\( 5^{n+2} = 5^n \cdot 5^2 = 25 \cdot 5^n \)[/tex]
- Similarly, [tex]\( 5^{n+1} = 5^n \cdot 5 = 5 \cdot 5^n \)[/tex]
So, the expression can be rewritten as:
[tex]\[
\frac{25 \cdot 5^n - 5^n}{5 \cdot 5^n + 5^n}
\][/tex]
2. Factor out the common term [tex]\(5^n\)[/tex] from both the numerator and the denominator:
The numerator becomes:
[tex]\[
25 \cdot 5^n - 5^n = 5^n(25 - 1)
\][/tex]
The denominator becomes:
[tex]\[
5 \cdot 5^n + 5^n = 5^n(5 + 1)
\][/tex]
So the expression now looks like:
[tex]\[
\frac{5^n(25 - 1)}{5^n(5 + 1)}
\][/tex]
3. Simplify the expressions inside the parentheses:
[tex]\[
25 - 1 = 24
\][/tex]
[tex]\[
5 + 1 = 6
\][/tex]
Now, the expression is:
[tex]\[
\frac{5^n \cdot 24}{5^n \cdot 6}
\][/tex]
4. Cancel out the common term [tex]\(5^n\)[/tex] from the numerator and the denominator:
[tex]\[
\frac{24}{6}
\][/tex]
5. Divide the numerator by the denominator:
[tex]\[
\frac{24}{6} = 4
\][/tex]
So, the simplified form of the given expression is:
[tex]\[
4
\][/tex]
Therefore, the final simplified expression is:
[tex]\[
\boxed{4}
\][/tex]
