To prove that:
[tex]\[
\frac{5^{m+2} - 5^m}{5^{m+1} + 5^m} = 4
\][/tex]
we can follow a step-by-step simplification process.
1. Rewrite the expression by factoring out common terms:
Since the terms in the numerator and the denominator involve powers of 5, we can factor out the lower power of 5 in each case.
[tex]\[
\frac{5^{m+2} - 5^m}{5^{m+1} + 5^m}
\][/tex]
2. Factor out [tex]\(5^m\)[/tex] in both the numerator and the denominator:
In the numerator:
[tex]\[
5^{m+2} = 5^m \cdot 5^2 = 25 \cdot 5^m
\][/tex]
Therefore, the numerator becomes:
[tex]\[
25 \cdot 5^m - 5^m = 5^m (25 - 1) = 5^m \cdot 24
\][/tex]
In the denominator:
[tex]\[
5^{m+1} = 5^m \cdot 5 = 5 \cdot 5^m
\][/tex]
Therefore, the denominator becomes:
[tex]\[
5 \cdot 5^m + 5^m = 5^m (5 + 1) = 5^m \cdot 6
\][/tex]
3. Reassemble the expression with the factored terms:
[tex]\[
\frac{5^m \cdot 24}{5^m \cdot 6}
\][/tex]
4. Simplify the expression by canceling out the common factor [tex]\(5^m\)[/tex]:
[tex]\[
\frac{24}{6} = 4
\][/tex]
Thus, we have shown that:
[tex]\[
\frac{5^{m+2} - 5^m}{5^{m+1} + 5^m} = 4
\][/tex]
This completes the proof.
