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Section 3.6 Laws of Exponents
Here’s a summary of the
Laws of Exponents encountered in this chapter. While it is important that you know and can apply these principles, avoid thinking of it just as a table of formulas to memorize. Rather, consider simple examples of each case and how we saw the expected result in the discussion throughout this chapter.
Table 3.6.1.
Zero Exponent:
\begin{equation*}
a^0 = 1, \quad a\neq 0
\end{equation*}
Unit Exponent:
\begin{equation*}
a^1 = a
\end{equation*}
Power of Product:
\begin{equation*}
(a b)^n = a^{n} b^n
\end{equation*}
Power of Quotient:
\begin{equation*}
\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
\end{equation*}
Product of Powers:
\begin{equation*}
a^m a^n = a^{m+n}
\end{equation*}
Quotient of Powers
\begin{equation*}
\frac{a^m}{a^n} = a^{m-n}
\end{equation*}
Reciprocal:
\begin{equation*}
a^{-1} = \frac{1}{a}
\end{equation*}
Negative Exponent:
\begin{equation*}
a^{-n} = \frac{1}{a^n}
\end{equation*}
Power of Power:
\begin{equation*}
(a^m)^n = a^{mn}
\end{equation*}
Rational Exponent:
\begin{equation*}
a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m
\end{equation*}
Example 3.6.2 . Using Laws of Exponents.
Simplify each expression into the form \(a^n\text{,}\) where you may assume \(a \gt 0\) is a positive real number.
\(\displaystyle \displaystyle a^5 \cdot a^3 \)
\(\displaystyle \displaystyle \frac{a^0}{a^{-5}}\)
\(\displaystyle \displaystyle \left(a^4 \cdot a^{-7}\right)^3\)
\(\displaystyle \displaystyle \frac{\sqrt[4]{a^5}}{\sqrt{a}}\)
Answer .
\(\displaystyle \displaystyle a^5 \cdot a^3 = a^8\)
\(\displaystyle \displaystyle \frac{a^0}{a^{-5}} =a^5 \)
\(\displaystyle \displaystyle \left(a^{-7} \cdot a^{4}\right)^3 = a^{-9}\)
\(\displaystyle \displaystyle \frac{\sqrt[4]{a^5}}{\sqrt{a}} = a^{3/4}\)
