Simplify :
`(5^(n+3)-6xx5^(n+1))/(9xx5^(n)-5^(n)xx2^(2))`

Answer

To simplify the expression \((5^{n+3} - 6 \times 5^{n+1}) / (9 \times 5^{n} - 5^{n} \times 2^2)\), we can follow these steps:

### Step 1: Rewrite the expression
We start by rewriting the expression:
\[ \frac{5^{n+3} - 6 \times 5^{n+1}}{9 \times 5^{n} - 5^{n} \times 2^2} \]

### Step 2: Factor out common terms in the numerator
In the numerator, we can factor out \(5^{n+1}\):
\[ = \frac{5^{n+1}(5^2 - 6)}{9 \times 5^{n} - 5^{n} \times 4} \]
This simplifies to:
\[ = \frac{5^{n+1}(25 - 6)}{9 \times 5^{n} - 4 \times 5^{n}} \]

### Step 3: Simplify the numerator
Now, simplify \(25 - 6\):
\[ = \frac{5^{n+1} \times 19}{9 \times 5^{n} - 4 \times 5^{n}} \]

### Step 4: Factor out common terms in the denominator
In the denominator, we can factor out \(5^{n}\):
\[ = \frac{5^{n+1} \times 19}{(9 - 4) \times 5^{n}} \]
This simplifies to:
\[ = \frac{5^{n+1} \times 19}{5 \times 5^{n}} \]

### Step 5: Cancel out \(5^{n}\)
Now, we can cancel \(5^{n}\) from the numerator and denominator:
\[ = \frac{19 \times 5^{n+1}}{5 \times 5^{n}} = \frac{19 \times 5^{n} \times 5}{5 \times 5^{n}} = \frac{19}{1} \]

### Step 6: Final Result
Thus, the simplified expression is:
\[ = 19 \]