Divide and simplify: \(\frac{4n}{n+5} \div \frac{n-9}{n+5}\)
Understand the Problem
The question asks to divide two rational expressions and simplify the result. That is, perform the operation (\frac{4n}{n+5} \div \frac{n-9}{n+5}) and simplify the answer.
Answer
$\frac{4n}{n-9}$
Answer for screen readers
$\frac{4n}{n-9}$
Steps to Solve
- Rewrite the division as multiplication by the reciprocal
Dividing by a fraction is the same as multiplying by its reciprocal. So we rewrite the expression as:
$$ \frac{4n}{n+5} \div \frac{n-9}{n+5} = \frac{4n}{n+5} \cdot \frac{n+5}{n-9} $$
- Multiply the fractions
Multiply the numerators and the denominators:
$$ \frac{4n}{n+5} \cdot \frac{n+5}{n-9} = \frac{4n(n+5)}{(n+5)(n-9)} $$
- Simplify by cancelling common factors
The term $(n+5)$ appears in both the numerator and the denominator, so we can cancel it out, provided that $n \neq -5$.
$$ \frac{4n(n+5)}{(n+5)(n-9)} = \frac{4n}{n-9} $$
$\frac{4n}{n-9}$
More Information
The expression is simplified as much as possible. There are no other common factors in the numerator and denominator. Also, $n$ cannot equal 9 since that would make the denominator zero, and $n$ cannot equal -5 since that would also make the original expression undefined.
Tips
A common mistake is to try to cancel terms that are added or subtracted. Only factors can be cancelled. For example, in the original expression, one can only cancel the common factor $(n+5)$.
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