Given something like:
$$\frac{a}{\frac{a}{b}}$$ You would multiply the numerator $a$ by the reciprocal of the denominator, $\tfrac ba$ to get: $$ a\cdot\frac{b}{a}= \frac{ab}{a}=b $$ Given $$ \frac{1}{\frac 1a + \frac 1b} $$ By taking the LCM and adding the denominators you get: $$ \frac{1}{\left(\frac{a+b}{ab}\right)} $$ Given the reciprocal division rule in example one: $$ \frac{ab}{a+b} $$
Why can you not take the reciprocal of $\frac 1a + \frac 1b$ to begin with? I did this and ended up with: $$ 1\cdot\left(\frac a1 + \frac b1\right) =a + b $$
However $\frac{1}{1/a + 1/b}$ is not the same as $a+b$ so this is incorrect. I was trying to find a similar example online but I could not, why is this incorrect? Does the rule only work with one fraction as the denominator and not terms linked by addition and/or subtraction?
When you took the reciprocal of $\frac 1a+\frac 1b$ to get $a+b$, you implicitly assumed that the reciprocal of a sum is the sum of the reciprocals. This is not true.
Note that $\frac 1a$ is $a^{-1}$. Just as the square of a sum is not the sum of the squares (in general), that is, $(x+y)^2\neq x^2+y^2$, so also the reciprocal of a sum is not in general the sum of the reciprocals: $(x+y)^{-1}\neq x^{-1}+y^{-1}$.