Simplifying Exponential Expression with Factoring Powers of 5

Math Problem Statement

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

We are tasked with simplifying the expression:

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

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

$5^{n+3} - 6(5^{n+1})$ Factor $5^{n+1}$ from both terms:

= 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.

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Exponential Functions
Factoring
Simplifying Expressions

a^m * a^n = a^(m+n)
Factoring out common terms

Laws of Exponents

Grades 10-12

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