- Convert the division of the two fractions into multiplication by the reciprocal: \(\frac{m+3}{m^2-16} \cdot \frac{m+4}{m^2-9}\).
- Factor the quadratic denominators: \((m^2-16) = (m-4)(m+4)\) and \((m^2-9) = (m-3)(m+3)\).
- Substitute the factors into the expression to get \(\frac{m+3}{(m-4)(m+4)} \cdot \frac{m+4}{(m-3)(m+3)}\).
- Cancel common factors \((m+3)\) and \((m+4)\) to reach the simplified result: \(\boxed{\frac{1}{(m-4)(m-3)}}\).
Explanation
- Analyzing and Rewriting the Expression
The given expression is a complex rational fraction:
\[\frac{\frac{m+3}{m^2-16}}{\frac{m^2-9}{m+4}}\]
To simplify this expression, we first recognize that dividing by a fraction is equivalent to multiplying by its reciprocal. Thus, we rewrite the expression as:
\[\frac{m+3}{m^2-16} \times \frac{m+4}{m^2-9}\]
We must also note the constraints: the denominators cannot be zero. This means \(m^2-16
eq 0\) (\(m
eq 4, -4\)), \(m+4
eq 0\) (\(m
eq -4\)), and \(m^2-9
eq 0\) (\(m
eq 3, -3\)). Thus, \(m
otin \{4, -4, 3, -3\}\).
- Factoring Quadratic Terms
Next, we factor the quadratic expressions in the denominators using the difference of squares identity, \(a^2 - b^2 = (a-b)(a+b)\):
- \(m^2 - 16 = (m-4)(m+4)\)
- \(m^2 - 9 = (m-3)(m+3)\)
Substituting these factored forms back into our expression gives:
\[\frac{m+3}{(m-4)(m+4)} \times \frac{m+4}{(m-3)(m+3)}\]
- Simplifying the Expression
Now, we group the terms in the numerator and the denominator:
\[\frac{(m+3)(m+4)}{(m-4)(m+4)(m-3)(m+3)}\]
We can cancel the common factors in the numerator and the denominator, which are \((m+3)\) and \((m+4)\), provided that \(m
eq -3\) and \(m
eq -4\):
\[\frac{1}{(m-4)(m-3)}\]
This result matches the second option.
Examples
Simplifying complex fractions is a fundamental skill in algebra that appears frequently when solving rational equations or analyzing physical systems. For example, in electrical engineering, when calculating total resistance of parallel circuits or analyzing signal transfer functions, we often end up with complex ratios. Knowing how to factor and cancel terms allows engineers to simplify these complex formulas into manageable equations, leading to quicker and more accurate designs.