Simplify 2/(1/a+1/b)

$\frac{2}{\frac{1}{a} + \frac{1}{b}}$

Step 1

Simplify the denominator.

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

$\frac{2}{\frac{1}{a} \cdot \frac{b}{b} + \frac{1}{b}}$

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

$\frac{2}{\frac{1}{a} \cdot \frac{b}{b} + \frac{1}{b} \cdot \frac{a}{a}}$

Write each expression with a common denominator of $a b$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{a}$ by $\frac{b}{b}$ .

$\frac{2}{\frac{b}{a b} + \frac{1}{b} \cdot \frac{a}{a}}$

Multiply $\frac{1}{b}$ by $\frac{a}{a}$ .

$\frac{2}{\frac{b}{a b} + \frac{a}{b a}}$

Reorder the factors of $b a$ .

$\frac{2}{\frac{b}{a b} + \frac{a}{a b}}$

Combine the numerators over the common denominator.

$\frac{2}{\frac{b + a}{a b}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$2 \frac{a b}{b + a}$

Step 3

Combine $2$ and $\frac{a b}{b + a}$ .

$\frac{2 a b}{b + a}$