To simplify the given expression [tex]\(\left(\frac{x}{x+2}\right)\left(\frac{x+2}{x+50}\right)\)[/tex] and determine which of the listed options is equivalent to it, let's follow these steps:
1. Write down the expression:
[tex]\[
\left(\frac{x}{x+2}\right)\left(\frac{x+2}{x+50}\right)
\][/tex]
2. Identify any common factors that can be canceled out:
- The numerator of the first fraction is [tex]\( x \)[/tex].
- The denominator of the first fraction is [tex]\( x + 2 \)[/tex].
- The numerator of the second fraction is [tex]\( x + 2 \)[/tex].
- The denominator of the second fraction is [tex]\( x + 50 \)[/tex].
3. Simplify by canceling out the common factors:
- Notice that the [tex]\( x + 2 \)[/tex] in the denominator of the first fraction and the [tex]\( x + 2 \)[/tex] in the numerator of the second fraction are common factors.
- We can cancel [tex]\( x + 2 \)[/tex] in the following way:
[tex]\[
\frac{x}{\cancel{x+2}} \cdot \frac{\cancel{x+2}}{x+50}
\][/tex]
4. After cancelling out the common factors, we are left with:
[tex]\[
\frac{x}{x+50}
\][/tex]
Hence, the given expression simplifies to [tex]\( \frac{x}{x+50} \)[/tex].
Compare this with the given options:
A. [tex]\( \frac{1}{50} \)[/tex]
B. [tex]\( \frac{x}{x+50} \)[/tex]
C. [tex]\( \frac{2x+2}{2x+52} \)[/tex]
D. [tex]\( x^2 + 2 \)[/tex]
The simplified expression matches option B:
[tex]\[ \frac{x}{x+50} \][/tex]
Thus, the equivalent expression is:
[tex]\[ \boxed{\frac{x}{x+50}} \][/tex]
