$\textit{Algebraic Expansion with Exponents}$
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 $(5^x-5^{-x})^2$.
\( \begin{align} \displaystyle
&= (5^x)^2-2 \times 5^x \times 5^{-x} + (5^{-x})^2 \\
&= 5^{2x}-2 \times 5^0 + 5^{-2x} \\
&= 5^{2x}-2 + \dfrac{1}{5^{2x}}
\end{align} \)
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