Simplify the rational expression.

StartStartFraction 2 d OverOver StartFraction d Over r 1 EndFraction plus StartFraction d Over r 2 EndFraction EndEndFraction2ddr1+dr2

We are tasked with simplifying the following rational expression:

\[
\frac{2d}{\frac{d}{r_1} + \frac{d}{r_2}}
\]

### Step-by-Step Simplification:

1. **Combine the denominators** inside the larger fraction. The denominator \(\frac{d}{r_1} + \frac{d}{r_2}\) can be simplified by factoring out \(d\):

   \[
   \frac{d}{r_1} + \frac{d}{r_2} = d \left( \frac{1}{r_1} + \frac{1}{r_2} \right)
   \]

2. **Substitute** this into the original expression:

   \[
   \frac{2d}{d \left( \frac{1}{r_1} + \frac{1}{r_2} \right)}
   \]

3. **Cancel out** the \(d\) from the numerator and denominator (assuming \(d \neq 0\)):

   \[
   \frac{2}{\frac{1}{r_1} + \frac{1}{r_2}}
   \]

4. **Simplify the denominator**. We need to combine the two fractions in the denominator:

   \[
   \frac{1}{r_1} + \frac{1}{r_2} = \frac{r_1 + r_2}{r_1 r_2}
   \]

5. **Rewrite the expression** as:

   \[
   \frac{2}{\frac{r_1 + r_2}{r_1 r_2}} = 2 \cdot \frac{r_1 r_2}{r_1 + r_2}
   \]

6. **Final simplified expression**:

   \[
   \frac{2r_1 r_2}{r_1 + r_2}
   \]

### Final Answer:

\[
\frac{2r_1 r_2}{r_1 + r_2}
\]

This is the simplified form of the given rational expression.

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Would you like more details on any step? Here are some related questions you might find helpful:

1. What is the meaning of a rational expression in mathematics?
2. How do you combine two fractions with different denominators?
3. Why is factoring important in simplifying expressions?
4. Can you explain the role of cancellation in rational expressions?
5. How would the simplification change if there were more terms in the denominator?

**Tip:** Always check for common factors between the numerator and denominator, as they can often be canceled out to simplify the expression further.

Simplify the rational expression: \(\frac{2d}{\frac{d}{r_1} + \frac{d}{r_2}}\)

Learn how to simplify the rational expression \(\frac{2d}{\frac{d}{r_1} + \frac{d}{r_2}}\) with a step-by-step solution. This problem involves key concepts in algebra, including simplifying complex fractions, combining fractions, and rational expression properties. Suitable for students in Grades 9-12.

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