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Evaluate the following expression:

[tex]\[
\frac{3 \times 27^{n+1} + 9 \times 3^{3n-1}}{8 \times 3^{3n} - 5 \times 27^n}
\][/tex]


Sagot :

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]