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 $x^{-\frac{1}{2}}(x^{\frac{3}{2}} + 2x^{\frac{1}{2}} – 3x^{-\frac{1}{2}})$.

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

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 $(5^x – 5^{-x})^2$.

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


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