- Identify the expression as a quotient of two fractions and convert it to a product: \(\frac{m+3}{m^2-16} \cdot \frac{m+4}{m^2-9}\).
- Factor the denominators using the difference of squares formula: \(\frac{m+3}{(m-4)(m+4)} \cdot \frac{m+4}{(m-3)(m+3)}\).
- Cancel the common factors \((m+3)\) and \((m+4)\) from both the numerator and the denominator.
- State the final simplified form as \(\boxed{\frac{1}{(m-4)(m-3)}}\).
Explanation
- Rewrite the expression using the division rule for fractions
To simplify the complex rational expression \(\frac{\frac{m+3}{m^2-16}}{\frac{m^2-9}{m+4}}\), we first identify the numerator and denominator fractions. The expression can be rewritten by multiplying the numerator by the reciprocal of the denominator:
\[ \frac{m+3}{m^2-16} \times \frac{m+4}{m^2-9} \]
This is based on the rule for dividing rational expressions: \(\frac{A/B}{C/D} = \frac{A}{B} \times \frac{D}{C}\).
- Factor the quadratic expressions
Next, we factor the quadratic expressions in the denominators using the difference of squares identity, which states that \(a^2 - b^2 = (a-b)(a+b)\):
\[ m^2 - 16 = (m-4)(m+4) \]
\[ m^2 - 9 = (m-3)(m+3) \]
Substituting these factored forms back into our expression, we get:
\[ \frac{m+3}{(m-4)(m+4)} \times \frac{m+4}{(m-3)(m+3)} \]
- Simplify by canceling common factors
Now, we combine the fractions into a single expression and identify common factors in the numerator and denominator that can be canceled:
\[ \frac{(m+3)(m+4)}{(m-4)(m+4)(m-3)(m+3)} \]
By canceling the common terms \((m+3)\) and \((m+4)\), we are left with:
\[ \frac{1}{(m-4)(m-3)} \]
This represents the simplified form of the original complex rational expression. Comparing this to the given options, it corresponds to the second choice.
Examples
In engineering and physics, complex formulas often involve ratios of ratios, such as when calculating effective resistance or relative velocities. Simplifying these expressions is a fundamental skill that allows for clearer analysis and more accurate modeling. For example, if you are determining how different mechanical parts scale together, you might express their proportions as rational functions, and reducing these expressions simplifies the calculation of the final output values.