Solve [5^n+3 - 6(5^n+1)]/[9(5^n) - 5^n(2^n)]

We are tasked with simplifying the expression:

\[
\frac{5^{n+3} - 6(5^{n+1})}{9(5^n) - 5^n(2^n)}
\]

### Step 1: Factor out powers of 5

First, let's rewrite the expression by factoring powers of \(5^n\) where possible.

#### Numerator:
\[
5^{n+3} - 6(5^{n+1})
\]
Factor \(5^{n+1}\) from both terms:

\[
5^{n+1}(5^2 - 6)
= 5^{n+1}(25 - 6)
= 5^{n+1}(19)
\]

So the numerator becomes:
\[
19 \cdot 5^{n+1}
\]

#### Denominator:
\[
9(5^n) - 5^n(2^n)
\]
Factor out \(5^n\):

\[
5^n(9 - 2^n)
\]

### Step 2: Simplify the expression

Now substitute the simplified forms of the numerator and denominator back into the expression:

\[
\frac{19 \cdot 5^{n+1}}{5^n(9 - 2^n)}
\]

### Step 3: Cancel common factors

We can cancel \(5^n\) from both the numerator and the denominator:

\[
\frac{19 \cdot 5}{9 - 2^n}
= \frac{95}{9 - 2^n}
\]

### Final Answer:
\[
\frac{95}{9 - 2^n}
\]

Would you like more details or have any other questions?

Here are some related questions you might find interesting:
1. What if \(2^n = 9\)? How does the expression simplify?
2. How would the expression behave as \(n\) approaches infinity?
3. What if the base was different, like \(3^n\) instead of \(5^n\)?
4. How would this expression change if an additional term was added in the numerator?
5. Can this expression have complex solutions for specific values of \(n\)?

**Tip:** When simplifying algebraic expressions involving exponents, always look for common factors to cancel before expanding terms further.

Simplify the complex exponential expression [5^(n+3) - 6(5^(n+1))] / [9(5^n) - 5^n(2^n)] by factoring powers of 5 and using laws of exponents. A step-by-step guide suitable for students in Grades 10-12.

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