Simplify (2d)/(d/r_1+d/r_2)

$\frac{2 d}{\frac{d}{r_{1}} + \frac{d}{r_{2}}}$

Step 1

Simplify the denominator.

To write $\frac{d}{r_{1}}$ as a fraction with a common denominator, multiply by $\frac{r_{2}}{r_{2}}$ .

$\frac{2 d}{\frac{d}{r_{1}} \cdot \frac{r_{2}}{r_{2}} + \frac{d}{r_{2}}}$

To write $\frac{d}{r_{2}}$ as a fraction with a common denominator, multiply by $\frac{r_{1}}{r_{1}}$ .

$\frac{2 d}{\frac{d}{r_{1}} \cdot \frac{r_{2}}{r_{2}} + \frac{d}{r_{2}} \cdot \frac{r_{1}}{r_{1}}}$

Write each expression with a common denominator of $r_{1} r_{2}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{d}{r_{1}}$ by $\frac{r_{2}}{r_{2}}$ .

$\frac{2 d}{\frac{d r_{2}}{r_{1} r_{2}} + \frac{d}{r_{2}} \cdot \frac{r_{1}}{r_{1}}}$

Multiply $\frac{d}{r_{2}}$ by $\frac{r_{1}}{r_{1}}$ .

$\frac{2 d}{\frac{d r_{2}}{r_{1} r_{2}} + \frac{d r_{1}}{r_{2} r_{1}}}$

Reorder the factors of $r_{2} r_{1}$ .

$\frac{2 d}{\frac{d r_{2}}{r_{1} r_{2}} + \frac{d r_{1}}{r_{1} r_{2}}}$

Combine the numerators over the common denominator.

$\frac{2 d}{\frac{d r_{2} + d r_{1}}{r_{1} r_{2}}}$

Factor $d$ out of $d r_{2} + d r_{1}$ .

Factor $d$ out of $d r_{2}$ .

$\frac{2 d}{\frac{d (r_{2}) + d r_{1}}{r_{1} (r_{2})}}$

Factor $d$ out of $d r_{1}$ .

$\frac{2 d}{\frac{d (r_{2}) + d (r_{1})}{r_{1} (r_{2})}}$

Factor $d$ out of $d (r_{2}) + d (r_{1})$ .

$\frac{2 d}{\frac{d (r_{2} + r_{1})}{r_{1} r_{2}}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$2 d \frac{r_{1} r_{2}}{d (r_{2} + r_{1})}$

Step 3

Cancel the common factor of $d$ .

Factor $d$ out of $2 d$ .

$d \cdot 2 \frac{r_{1} r_{2}}{d (r_{2} + r_{1})}$

Cancel the common factor.

$d \cdot 2 \frac{r_{1} r_{2}}{d (r_{2} + r_{1})}$

Rewrite the expression.

$2 \frac{r_{1} r_{2}}{r_{2} + r_{1}}$

Step 4

Combine $2$ and $\frac{r_{1} r_{2}}{r_{2} + r_{1}}$ .

$\frac{2 r_{1} r_{2}}{r_{2} + r_{1}}$