- Convert the division by a fraction into multiplication by its reciprocal.
- Factor all numerators and denominators: \(3x+18=3(x+6)\), \(4x+8=4(x+2)\), and \(x^2-36=(x-6)(x+6)\).
- Cancel out the common factors \((x+6)\) and \((x+2)\).
- The simplified expression is \(\boxed{\frac{3}{4x-24}}\).
Explanation
- Rewriting Division as Multiplication
To simplify the expression \(\frac{3x+18}{4x+8} \div \frac{x^2-36}{x+2}\), we begin by recalling that dividing by a fraction is equivalent to multiplying by its reciprocal. Therefore, we rewrite the problem as:
\[\frac{3x+18}{4x+8} \times \frac{x+2}{x^2-36}\]
- Factoring the Expressions
Next, we factor each polynomial involved:
- The numerator \(3x+18\) can be factored as \(3(x+6)\).
- The denominator \(4x+8\) can be factored as \(4(x+2)\).
- The denominator \(x^2-36\) is a difference of squares, which factors as \((x-6)(x+6)\).
Substituting these factored forms back into our expression gives:
\[\frac{3(x+6)}{4(x+2)} \times \frac{x+2}{(x-6)(x+6)}\]
- Simplifying by Canceling Common Factors
Now we look for common factors in the numerator and the denominator that can be canceled:
- The term \((x+6)\) appears in both the numerator and the denominator, so we can cancel it out.
- The term \((x+2)\) also appears in both the numerator and the denominator, so we can cancel it as well.
After canceling these factors, the expression simplifies to:
\[\frac{3}{4(x-6)}\]
Distributing the 4 in the denominator, we get:
\[\frac{3}{4x-24}\]
- Conclusion
Comparing our simplified expression \(\frac{3}{4x-24}\) with the given options, we identify that it matches the third option:
◯ \(\frac{4(x-6)}{3}\)
◯ \(\frac{3}{4x-8}\)
◯ \(\frac{3}{4x-24}\)
◯ \(\frac{1}{4(x-2)}\)
Thus, the correct choice is \(\frac{3}{4x-24}\).
Examples
Simplifying rational expressions is essential in fields like physics and engineering, where complex formulas need to be reduced to manageable forms to make predictions. For example, when calculating the effective resistance of an electrical circuit with parallel components, one often encounters rational expressions. Simplifying these algebraic expressions allows engineers to quickly determine how changing the value of a single resistor will affect the overall behavior of the entire system.