How to shorten this fraction? How to shorten this fraction?
$R_1+R_2$ divided by $\frac1{R_1} + \frac1{R_2}$
The answer is $R_1R_2$. I just don't know how to get there.
 A: First, simplify the denominator:
$$\frac{1}{A}+\frac{1}{B} = \frac{B+A}{AB},$$
and now we can simplify the whole "castle":
$$\frac{A+B}{\frac{1}{A}+\frac{1}{B}} = \frac{A+B}{\frac{A+B}{AB}}= \frac{AB}{A+B}\cdot \frac{A+B}{1} = AB.$$
A: We can use a common trick for simplifying a fraction, multiplying by an expression equal to 1:
$$\frac{R_1+R_2}{\frac{1}{R_1}+\frac{1}{R_2}}=\frac{R_1+R_2}{\frac{1}{R_1}+\frac{1}{R_2}}\cdot\left(\frac{R_1R_2}{R_1R_2}\right)=\frac{(R_1+R_2)(R_1R_2)}{\left(\frac{1}{R_1}+\frac{1}{R_2}\right)(R_1R_2)}=\frac{(R_1+R_2)(R_1R_2)}{R_2+R_1}=R_1R_2$$
A: Multiply through by $R_1R_2$:
$$\frac{R_1+R_2}{1/R_1+1/R_2}=\frac{R_1R_2(R_1+R_2)}{R_1R_2(1/R_1+1/R_2)}=\frac{R_1R_2(R_1+R_2)}{R_2+R_1}=R_1R_2$$
A: Write
$$
\frac{R_1+R_2}{\frac{1}{R_1}+\frac{1}{R_2}}
$$
Multiply both numerator and denominator by $R_1R_2$ and cancel the $R_1+R_2$ in the numerator and denominator.
A: $R_1 + R_2$ divided by $\frac{1}{R_1} + \frac{1}{R_2}$
= $R_1 + R_2$ divided by $\frac{R_2}{R_1 R_2} + \frac{R_1}{R_1 R_2}$
= $R_1 + R_2$ divided by $\frac{R_2 + R_1}{R_1 R_2}$
= $R_1 + R_2$ times $\frac{R_1 R_2}{R_2 + R_1}$
= $\frac{R_1 + R_2}{1} \cdot \frac{R_1 R_2}{R_2 + R_1} = \frac{1}{1} \cdot \frac{R_1 R_2}{1}$
$= R_1R_2$
