3.6 Negative exponents

$\begin{array}{ll}4\cdot \frac{1}{4}=1.\hfill & \text{This}\text{means}\text{that}4\text{and}\frac{1}{4}\text{are}\text{reciprocals}.\hfill \end{array}$

$\begin{array}{ll}6\cdot \frac{1}{6}=1.\hfill & \text{Hence,}6\text{and}\frac{1}{6}\text{are}\text{reciprocals}.\hfill \end{array}$

$\begin{array}{ll}-2\cdot \frac{-1}{2}=1.\hfill & \text{Hence,}-2\text{and}-\frac{1}{2}\text{are}\text{reciprocals}.\hfill \end{array}$

$\begin{array}{ll}a\cdot \frac{1}{a}=1.\hfill & \text{Hence,}a\text{and}\frac{1}{a}\text{are}\text{reciprocals}\text{if}a\ne 0.\hfill \end{array}$

$\begin{array}{ll}x\cdot \frac{1}{x}=1.\hfill & \text{Hence,}x\text{and}\frac{1}{x}\text{are}\text{reciprocals}\text{if}x\ne 0.\hfill \end{array}$

$\begin{array}{ll}{x}^3\cdot \frac{1}{x^3}=1.\hfill & \text{Hence,}{x}^3\text{and}\frac{1}{x^3}\text{are}\text{reciprocals}\text{if}x\ne 0.\hfill \end{array}$

$x^{-6}=\frac{1}{x^6}$

$a^{-1}=\frac{1}{a^1}=\frac{1}{a}$

$7^{-2}=\frac{1}{7^2}=\frac{1}{49}$

${(3a)}^{-6}=\frac{1}{{(3a)}^6}$

${(5x-1)}^{-24}=\frac{1}{{(5x-1)}^{24}}$

${(k+2z)}^{-(-8)}={(k+2z)}^8$

$\begin{array}{ll}{x}^{-2}{y}^5.\hfill & \text{The}factor{x}^{-2}\text{can}\text{be}\text{moved}\text{from}\text{the}\text{numerator}\text{to}\text{the}\hfill \\ \hfill & \text{denominator}\text{by}\text{changing}\text{the}\text{exponent}-2\text{to}+2.\hfill \\ x^{-2}{y}^5=\frac{y^5}{x^2}\hfill & \hfill \end{array}$

$\begin{array}{ll}{a}^9{b}^{-3}.\hfill & \text{The}factor{b}^{-3}\text{can}\text{be}\text{moved}\text{from}\text{the}\text{numerator}\text{to}\text{the}\hfill \\ \hfill & \text{denominator}\text{by}\text{changing}\text{the}\text{exponent}-3\text{to}+3.\hfill \\ a^9{b}^{-3}=\frac{a^9}{b^3}\hfill & \hfill \end{array}$

$\begin{array}{ll}\frac{a^4{b}^2}{c^{-6}}.\hfill & \text{This}\text{fraction}\text{can}\text{be}\text{written}\text{without}\text{any}\text{negative}\text{exponents}\hfill \\ \hfill & \text{by}\text{moving}\text{the}factor{c}^{-6}\text{into}\text{the}\text{numerator}\text{.}\hfill \\ \hfill & \text{We}\text{must}\text{change}\text{the}-6\text{to}+6\text{to}\text{make}\text{the}\text{move}\text{legitimate}\text{.}\hfill \\ \frac{a^4{b}^2}{c^{-6}}=a^4{b}^2{c}^6\hfill & \hfill \end{array}$