Algebraic Factorisation with Exponents (Indices)

$\textit{Factorisation}$

We first look for $\textit{common factors}$ and then for other forms such as $\textit{perfect squares}$, $\textit{difference of two squares}$, etc.

Example 1

Factorise $2^{n+4} + 2^{n+1}$.

Example 2

Factorise $2^{n+3} + 16$.
We can use the difference of squares rule: $$a^2 - b^2 = (a-b)(a+b)$$

Example 3

Factorise $9^x - 16$.
We can use the perferct square rules: $$a^2 + 2ab + b^2 = (a+b)^2$$ $$a^2 - 2ab + b^2 = (a-b)^2$$

Example 4

Factorise $9^x + 4 \times 3^x + 4$.

Example 5

Factorise $4^x - 2^x - 6$.

Example 6

Factorise $3^{2x+1} + 6 \times 3^x - 24$.