Let's work out the given expression step-by-step.
Given:
[tex]\[
\frac{5^2 \cdot 2^n + 2^{n+1} - 3^2 \cdot 2^n}{3^{m+3} - 2^2 \cdot 3^{m+1}}
\][/tex]
### Numerator Calculation:
1. Compute [tex]\(5^2\)[/tex]:
[tex]\[
5^2 = 25
\][/tex]
2. Compute [tex]\(3^2\)[/tex]:
[tex]\[
3^2 = 9
\][/tex]
3. The numerator [tex]\(25 \cdot 2^n + 2^{n+1} - 9 \cdot 2^n\)[/tex]:
- Combine like terms: [tex]\(25 \cdot 2^n - 9 \cdot 2^n\)[/tex]:
[tex]\[
(25 - 9) \cdot 2^n = 16 \cdot 2^n
\][/tex]
- Add the remaining term [tex]\(2^{n+1}\)[/tex]:
[tex]\[
16 \cdot 2^n + 2^{n+1}
\][/tex]
4. Observe that [tex]\(2^{n+1} = 2 \cdot 2^n\)[/tex], so rewrite the numerator:
[tex]\[
16 \cdot 2^n + 2 \cdot 2^n = (16 + 2) \cdot 2^n = 18 \cdot 2^n
\][/tex]
### Denominator Calculation:
1. Work on the term [tex]\(3^{m+3}\)[/tex]:
By the properties of exponents:
[tex]\[
3^{m+3} = 3^m \cdot 3^3 = 3^m \cdot 27
\][/tex]
2. Work on the term [tex]\(2^2 \cdot 3^{m+1}\)[/tex]:
First, calculate [tex]\(2^2\)[/tex]:
[tex]\[
2^2 = 4
\][/tex]
Now rewrite the term using exponent rules:
[tex]\[
4 \cdot 3^{m+1} = 4 \cdot 3^m \cdot 3 = 12 \cdot 3^m
\][/tex]
3. The denominator becomes:
[tex]\[
3^m \cdot 27 - 12 \cdot 3^m
\][/tex]
4. Factor out [tex]\(3^m\)[/tex]:
[tex]\[
3^m (27 - 12) = 3^m \cdot 15
\][/tex]
### Final Expression:
Putting both the numerator and the denominator together, we get:
[tex]\[
\frac{18 \cdot 2^n}{15 \cdot 3^m}
\][/tex]
### Simplification:
Simplify the coefficient:
[tex]\[
\frac{18}{15} \cdot \frac{2^n}{3^m} = \frac{6}{5} \cdot \frac{2^n}{3^m}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\frac{6 \cdot 2^n}{5 \cdot 3^m}
\][/tex]
This is the final result obtained by simplifying the given mathematical expression.
