# 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:

\large \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 $x^{-\frac{1}{2}}\left(x^{\frac{3}{2}} + 2x^{\frac{1}{2}}-3x^{-\frac{1}{2}}\right)$.

\begin{align} \displaystyle &= x^{-\frac{1}{2}} \times x^{\frac{3}{2}} + x^{-\frac{1}{2}} \times 2x^{\frac{1}{2}}-x^{-\frac{1}{2}} \times 3x^{-\frac{1}{2}} \\ &= x^{-\frac{1}{2} + \frac{3}{2}} + 2x^{-\frac{1}{2} + \frac{1}{2}} -3x^{-\frac{1}{2}-\frac{1}{2}} \\ &= x^1 + 2x^0-3x^{-1} \\ &= x + 2-\dfrac{3}{x} \end{align}

### Example 2

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

\begin{align} \displaystyle &= 2^x \times 2^x-2^x \times 2 + 2^x-2 \\ &= 2^{2x}-2^{x+1} + 2^x-2 \end{align}

### Example 3

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

\begin{align} \displaystyle &= 4^x \times 8^x-4^x \times 2 + 2 \times 8^x-4 \\ &= (2^2)^x \times (2^3)^x-(2^2)^x \times 2 + 2 \times (2^3)^x-4 \\ &= 2^{2x} \times 2^{3x}-2^{2x} \times 2 + 2 \times 2^{3x}-4 \\ &= 2^{5x}-2^{2x+1} + 2^{3x+1}-4 \end{align}

### Example 4

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

\begin{align} \displaystyle &= (5^x)^2-2 \times 5^x \times 5^{-x} + (5^{-x})^2 \\ &= 5^{2x}-2 \times 5^0 + 5^{-2x} \\ &= 5^{2x}-2 + \dfrac{1}{5^{2x}} \end{align}