Answer :
To solve the given expression [tex]\(\frac{7 \cdot 3^{n+3}}{3^{n+4} - 6 \cdot 3^{n+1}}\)[/tex], we will simplify it step by step.
### Step 1: Rewrite Exponents
First, let's rewrite the exponents in a way that we can better see opportunities for simplification.
- [tex]\(3^{n+4}\)[/tex] can be rewritten as [tex]\(3 \cdot 3^{n+3}\)[/tex].
- [tex]\(6 \cdot 3^{n+1}\)[/tex] can be rewritten as [tex]\(6 \cdot 3 \cdot 3^n\)[/tex] or [tex]\(2 \cdot 3^2 \cdot 3^n\)[/tex] which becomes [tex]\(2 \cdot 3^{n+2}\)[/tex].
Now the expression looks like this:
[tex]\[ \frac{7 \cdot 3^{n+3}}{3 \cdot 3^{n+3} - 2 \cdot 3^{n+2}} \][/tex]
### Step 2: Factor the Denominator
Next, we can factor [tex]\(3^{n+2}\)[/tex] from the terms in the denominator:
[tex]\[ 3 \cdot 3^{n+3} = 3^{n+2} \cdot 3^2 \][/tex]
So the expression in the denominator becomes:
[tex]\[ 3^{n+2} \cdot 3 - 2 \cdot 3^{n+2} \][/tex]
Factoring out [tex]\(3^{n+2}\)[/tex] gives:
[tex]\[ 3^{n+2}(3 - 2) \][/tex]
Simplifying inside the parentheses:
[tex]\[ 3 - 2 = 1 \][/tex]
Therefore, the denominator simplifies to:
[tex]\[ 3^{n+2} \cdot 1 = 3^{n+2} \][/tex]
### Step 3: Simplify the Fraction
Now the simplified fraction is:
[tex]\[ \frac{7 \cdot 3^{n+3}}{3^{n+2}} \][/tex]
We can simplify the exponents in the numerator and the denominator:
[tex]\[ \frac{7 \cdot 3^{n+3}}{3^{n+2}} = 7 \cdot 3^{(n+3) - (n+2)} = 7 \cdot 3^{1} = 7 \cdot 3 \][/tex]
### Step 4: Final Simplification
Multiplying the constants gives:
[tex]\[ 7 \cdot 3 = 21 \][/tex]
### Conclusion
The simplified result is:
[tex]\[ \boxed{21} \][/tex]
### Step 1: Rewrite Exponents
First, let's rewrite the exponents in a way that we can better see opportunities for simplification.
- [tex]\(3^{n+4}\)[/tex] can be rewritten as [tex]\(3 \cdot 3^{n+3}\)[/tex].
- [tex]\(6 \cdot 3^{n+1}\)[/tex] can be rewritten as [tex]\(6 \cdot 3 \cdot 3^n\)[/tex] or [tex]\(2 \cdot 3^2 \cdot 3^n\)[/tex] which becomes [tex]\(2 \cdot 3^{n+2}\)[/tex].
Now the expression looks like this:
[tex]\[ \frac{7 \cdot 3^{n+3}}{3 \cdot 3^{n+3} - 2 \cdot 3^{n+2}} \][/tex]
### Step 2: Factor the Denominator
Next, we can factor [tex]\(3^{n+2}\)[/tex] from the terms in the denominator:
[tex]\[ 3 \cdot 3^{n+3} = 3^{n+2} \cdot 3^2 \][/tex]
So the expression in the denominator becomes:
[tex]\[ 3^{n+2} \cdot 3 - 2 \cdot 3^{n+2} \][/tex]
Factoring out [tex]\(3^{n+2}\)[/tex] gives:
[tex]\[ 3^{n+2}(3 - 2) \][/tex]
Simplifying inside the parentheses:
[tex]\[ 3 - 2 = 1 \][/tex]
Therefore, the denominator simplifies to:
[tex]\[ 3^{n+2} \cdot 1 = 3^{n+2} \][/tex]
### Step 3: Simplify the Fraction
Now the simplified fraction is:
[tex]\[ \frac{7 \cdot 3^{n+3}}{3^{n+2}} \][/tex]
We can simplify the exponents in the numerator and the denominator:
[tex]\[ \frac{7 \cdot 3^{n+3}}{3^{n+2}} = 7 \cdot 3^{(n+3) - (n+2)} = 7 \cdot 3^{1} = 7 \cdot 3 \][/tex]
### Step 4: Final Simplification
Multiplying the constants gives:
[tex]\[ 7 \cdot 3 = 21 \][/tex]
### Conclusion
The simplified result is:
[tex]\[ \boxed{21} \][/tex]
