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Section 3.7 Exponential Expressions
Recall that
\begin{equation*}
a^{-n} = \frac{1}{a}
\end{equation*}
and
\begin{equation*}
a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m.
\end{equation*}
(See
Chapter 2.) Calculus will require frequently moving between these exponential and algebraic forms.
Example 3.28. Simplifying an Exponential Expression.
Assume \(x\) and \(y\) are non-zero real numbers.
\begin{equation*}
\left[ \left( \frac{-2x^{-1}y^2}{x^4 y^0} \right)^3 \right]^2.
\end{equation*}
Solution.
One possible approach:
\begin{align*}
\left[\left(\frac{-2x^{-1}y^2}{x^4 y^0}\right)^3\right]^2
&= \left(\frac{-2x^{-1}y^2}{x^4 y^0}\right)^6\\
&= \left(\frac{-2y^2}{x \cdot x^4 \cdot 1}\right)^6\\
&= \left(\frac{-2y^2}{x^5}\right)^6\\
&= \frac{(-2)^6(y^2)^6}{(x^5)^6}\\
&= \frac{64 y^{12}}{x^{30}}
\end{align*}
Example 3.29. Simplifying an Exponential Expression.
Simplify the expression completely. Assume \(x\) and \(y\) are positive real numbers.
\begin{equation*}
\sqrt{9x^2 y^5} \sqrt{16 x^{-3} y}
\end{equation*}
Solution.
One possible approach is to express in terms of rational exponents and use the Laws of Exponents we’ve discussed.
\begin{align*}
\sqrt{9x^2 y^5} \sqrt{16 x^{-3} y}
&= \left(9x^2 y^5\right)^{1/2} \left(16 x^{-3} y\right)^{1/2}\\
&= 3 x^1 y^{5/2} \cdot 4 x^{-3/2} y^{1/2}\\
&= 12 x^{1-3/2} y^{5/2+1/2}\\
&= 12 x^{-1/2} y^{3}\\
&= \frac{12 y^3}{x^{1/2}}\\
&= \frac{12 y^3}{\sqrt{x}}
\end{align*}
Example 3.30. Simplifying an Exponential Expression.
Rewrite the expression in exponential notation using negative or rational powers of \(x\) where appropriate.
\begin{equation*}
4x^3 +\frac{2}{x^5} - \frac{1}{2x}+ \frac{3}{5 \sqrt{x}} + 2 \sqrt[4]{x^3} - 5
\end{equation*}
Solution.
\begin{gather*}
4x^3 +\frac{2}{x^5} - \frac{1}{2x}+ \frac{3}{5 \sqrt{x}} + 2 \sqrt[4]{x^3} - 5\\
=
4x^3 +2x^{-5} - \frac{1}{2} x^{-1}+ \frac{3}{5} x^{-1/2} + 2 x^{3/4} - 5
\end{gather*}
