Algebraic Expansion with Exponents (Indices)


$\textit{Algebraic Expansion with Exponents}$

Expansion of algebraic expressions like $x^{\frac{1}{3}}(4x^{\frac{4}{5}} - 3x^{\frac{3}{2}})$, $(4x^5 + 6)(5^x - 7)$ and $(4^x + 7)^2$ are handled in the same way, using the same expansion laws to simplify expressions containing exponents:

$$ \begin{align} \displaystyle
a(a+b) &= ab+ac \\
(a+b)(c+d) &= ac+ad+bc+bd \\
(a+b)(a-b) &= a^2 -b^2 \\
(a+b)^2 &= a^2 + 2ab + b^2 \\
(a-b)^2 &= a^2 - 2ab + b^2 \\
\end{align} $$

Example 1

Expand and simplify $x^{-\frac{1}{2}}(x^{\frac{3}{2}} + 2x^{\frac{1}{2}} - 3x^{-\frac{1}{2}})$.

Example 2

Expand and simplify $(2^x + 1)(2^x - 2)$.

Example 3

Expand and simplify $(4^x + 2)(8^x - 2)$.

Example 4

Expand and simplify $(5^x - 5^{-x})^2$.