Certainly! Let's evaluate the given fraction step-by-step.
We are given the expression:
[tex]\[
\frac{3 \times 27^{n+1} + 9 \times 3^{3n-1}}{8 \times 3^{3n} - 5 \times 27^n}
\][/tex]
### Step 1: Simplify the Numerator
The numerator is:
[tex]\[
3 \times 27^{n+1} + 9 \times 3^{3n-1}
\][/tex]
First, we will rewrite [tex]\(27^{n+1}\)[/tex] in terms of base 3:
[tex]\[
27 = 3^3 \implies 27^{n+1} = (3^3)^{n+1} = 3^{3(n+1)} = 3^{3n + 3}
\][/tex]
Now substitute this back into the numerator:
[tex]\[
3 \times 3^{3n + 3} + 9 \times 3^{3n-1}
\][/tex]
Since [tex]\(9 = 3^2\)[/tex], we can also rewrite [tex]\(9 \times 3^{3n-1}\)[/tex]:
[tex]\[
9 \times 3^{3n-1} = 3^2 \times 3^{3n-1} = 3^{2 + 3n - 1} = 3^{3n + 1}
\][/tex]
Now our numerator becomes:
[tex]\[
3 \times 3^{3n + 3} + 3^{3n + 1}
\][/tex]
Combine the terms:
[tex]\[
3^{3n + 4} + 3^{3n + 1}
\][/tex]
Factor out the common term [tex]\(3^{3n + 1}\)[/tex]:
[tex]\[
3^{3n + 1} (3^3 + 1) = 3^{3n + 1} (27 + 1) = 3^{3n + 1} \times 28
\][/tex]
Thus, the numerator simplifies to:
[tex]\[
84 \times 3^{3n}
\][/tex]
### Step 2: Simplify the Denominator
The denominator is:
[tex]\[
8 \times 3^{3n} - 5 \times 27^n
\][/tex]
Rewrite [tex]\(27^n\)[/tex] in terms of base 3:
[tex]\[
27 = 3^3 \implies 27^n = (3^3)^n = 3^{3n}
\][/tex]
Now substitute this back into the denominator:
[tex]\[
8 \times 3^{3n} - 5 \times 3^{3n}
\][/tex]
Combine terms:
[tex]\[
(8 - 5) \times 3^{3n} = 3 \times 3^{3n}
\][/tex]
So, the denominator simplifies to:
[tex]\[
3 \times 3^{3n} = 3^{3n + 1}
\][/tex]
### Step 3: Simplify the Fraction
Now, we have the simplified form of the numerator and the denominator:
[tex]\[
\frac{84 \times 3^{3n}}{3^{3n + 1}}
\][/tex]
We can now simplify this fraction by canceling out [tex]\(3^{3n}\)[/tex] in the numerator and denominator:
[tex]\[
\frac{84 \times 3^{3n}}{3 \times 3^{3n}} = \frac{84}{3} = 28
\][/tex]
Thus, the simplified value of the given expression is:
[tex]\[
28
\][/tex]
