Question Type 2: Proving starting from the right-hand side Exercises

Starting from the left-hand side, prove that $\dfrac{1}{\frac{1}{a} + \frac{1}{b}} = \dfrac{ab}{a+b}$.

Starting from the right-hand side, prove that $a^m \cdot a^n = a^{m+n}$ where $a \neq 0$ and $m, n$ are positive integers.

Starting from the right-hand side, prove that $\frac{1}{a} + \frac{1}{b} = \frac{a+b}{ab}$.

Starting from the right-hand side, prove that $\log_b a = \dfrac{\ln a}{\ln b}$ for $a, b > 0$, $b \neq 1$.

Starting from the right-hand side, prove that $1 + r + r^2 + \dots + r^n = \dfrac{r^{n+1} - 1}{r - 1}$ for $r\neq 1$.

Starting from the right-hand side, prove that $(a+b)^2 = a^2 + 2ab + b^2$.