Multiply.
$\frac{4(x+6)}{(x-2)} \cdot \frac{(x+10)}{24(x+6)}$
Enter your answer, in simplest form, in the box.
We are given the expression \(\frac{4(x+6)}{(x-2)} \cdot \frac{(x+10)}{24(x+6)}\). Our objective is to simplify this product of two rational functions. First, we identify that the expression involves algebraic terms that can be canceled. We also note that for the expression to be defined, the denominators must not be zero. Thus, we have the restrictions \(x-2 eq 0\) (which means \(x eq 2\)) and \(x+6 eq 0\) (which means \(x eq -6\)).
To simplify, we multiply the two fractions together into one:
\[\frac{4(x+6)}{(x-2)} \cdot \frac{(x+10)}{24(x+6)} = \frac{4(x+6)(x+10)}{24(x-2)(x+6)}\]
By looking at the numerator and denominator, we can clearly see a common factor of \((x+6)\). Assuming \(x
eq -6\), we can cancel this factor from both the numerator and the denominator, which simplifies the expression significantly:
\[\frac{4(x+6)(x+10)}{24(x-2)(x+6)} = \frac{4(x+10)}{24(x-2)}\]
Now, we look at the numerical coefficients in the remaining fraction: \(\frac{4}{24}\). We can divide both the numerator and the denominator by their greatest common divisor, which is \(4\):
\[\frac{4}{24} = \frac{4 \div 4}{24 \div 4} = \frac{1}{6}\]
Substituting this back into the expression, we get:
\[\frac{1 \cdot (x+10)}{6 \cdot (x-2)} = \frac{x+10}{6(x-2)}\]
Thus, the expression in its simplest form is \(\frac{x+10}{6(x-2)}\).
Simplifying rational expressions is a fundamental skill in algebra with many practical applications. For instance, in engineering or physics, when you are modeling a physical system, you might end up with complex ratios representing ratios of forces, resistances, or velocities. Simplifying these expressions allows you to analyze how the system behaves as a whole, identify key factors that drive the system, and perform further calculations more efficiently. By learning to recognize and cancel common factors, you are essentially learning how to streamline complex mathematical descriptions into manageable forms.