Sure, let's simplify the expression [tex]\(\frac{7^{n+2} + 4 \times 7^n}{7^{n+1} + 8 - 3 \times 7^n}\)[/tex] step-by-step.
1. Numerator: [tex]\(7^{n+2} + 4 \times 7^n\)[/tex]
Rewrite [tex]\(7^{n+2}\)[/tex] as [tex]\(7^2 \times 7^n\)[/tex]:
[tex]\[
7^{n+2} = 49 \times 7^n
\][/tex]
Therefore, the numerator becomes:
[tex]\[
49 \times 7^n + 4 \times 7^n = (49 + 4) \times 7^n = 53 \times 7^n
\][/tex]
2. Denominator: [tex]\(7^{n+1} + 8 - 3 \times 7^n\)[/tex]
Rewrite [tex]\(7^{n+1}\)[/tex] as [tex]\(7 \times 7^n\)[/tex]:
[tex]\[
7^{n+1} = 7 \times 7^n
\][/tex]
So, the expression in the denominator becomes:
[tex]\[
7 \times 7^n + 8 - 3 \times 7^n
\][/tex]
Combine the terms involving [tex]\(7^n\)[/tex]:
[tex]\[
7 \times 7^n - 3 \times 7^n + 8 = (7 - 3) \times 7^n + 8 = 4 \times 7^n + 8
\][/tex]
3. Simplify the fraction:
[tex]\[
\frac{53 \times 7^n}{4 \times 7^n + 8}
\][/tex]
We can factor out [tex]\(7^n\)[/tex] in the denominator:
[tex]\[
\frac{53 \times 7^n}{4 \times 7^n + 8} = \frac{53 \times 7^n}{4(7^n + 2)}
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\frac{53 \times 7^n}{4 \times (7^n + 2)}
\][/tex]
