- Recognize the expression as a division of rational functions: \(\frac{m+3}{m^2-16} \cdot \frac{m+4}{m^2-9}\).
- Factor the quadratic expressions: \(m^2-16=(m-4)(m+4)\) and \(m^2-9=(m-3)(m+3)\).
- Multiply the fractions and cancel the common factors \((m+3)\) and \((m+4)\).
- Obtain the simplified expression: \(\boxed{\frac{1}{(m-4)(m-3)}}\).
Explanation
- Analyzing the Expression
To simplify the given complex rational expression, we first analyze the structure of the fraction:
\[ \frac{\frac{m+3}{m^2-16}}{\frac{m^2-9}{m+4}} \]
The expression is a division of two rational expressions, which can be rewritten as the product of the first expression and the reciprocal of the second:
\[ \frac{m+3}{m^2-16} \cdot \frac{m+4}{m^2-9} \]
We must also note the domain restrictions where the denominators cannot be zero: \(m^2-16
eq 0 \implies m
eq \pm 4\), \(m^2-9
eq 0 \implies m
eq \pm 3\), and \(m+4
eq 0 \implies m
eq -4\). Thus, \(m
otin \{ -4, -3, 3, 4 \}\).
- Factoring the Expressions
Next, we factor the quadratic expressions in the denominators and numerator:
- \(m^2 - 16\) is a difference of squares, so \(m^2 - 16 = (m-4)(m+4)\).
- \(m^2 - 9\) is also a difference of squares, so \(m^2 - 9 = (m-3)(m+3)\).
Substituting these factored forms back into our expression gives:
\[ \frac{m+3}{(m-4)(m+4)} \cdot \frac{m+4}{(m-3)(m+3)} \]
- Simplifying by Canceling Common Terms
Now, we multiply the two rational expressions together:
\[ \frac{(m+3) \cdot (m+4)}{(m-4)(m+4) \cdot (m-3)(m+3)} \]
We can cancel the common factors in the numerator and denominator, namely \((m+3)\) and \((m+4)\), provided that these factors are non-zero:
\[ \frac{1 \cdot 1}{(m-4) \cdot (m-3)} = \frac{1}{(m-4)(m-3)} \]
This simplified result matches the second given option.
Examples
Rational expressions appear frequently in engineering and physics, such as when calculating the combined resistance of parallel electrical circuits or finding the average speed over a journey with varying segments. For instance, if you have a complex formula for signal transmission efficiency involving multiple ratios, you can use these simplification techniques to reduce the formula to a more manageable form. This process not only saves time during calculations but also makes it easier to spot patterns and trends in the behavior of the system, helping engineers optimize their designs efficiently.