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Section 3.4 Summary of the Rules of Exponents

  • The Product Rule for Exponents:

    \begin{equation*} x^m \cdot x^n=x^{m+n} \end{equation*}

  • The Power to a Power Rule for Exponents:

    \begin{equation*} (x^m)^n=x^{mn} \end{equation*}

  • The Product to a Power Rule for Exponents:

    \begin{equation*} (xy)^n=x^ny^n \end{equation*}

  • The Quotient to a Power Rule for Exponents:

    \begin{equation*} \left(\frac{x}{y}\right)^n=\frac{x^n}{y^n} \end{equation*}

  • The Quotient Rules for Exponents:

    \begin{equation*} \frac{x^m}{x^n}=x^{m-n} \text{ and } \frac{x^m}{x^n}=\frac{1}{x^{n-m}} \end{equation*}

  • Zero Exponent:

    \begin{equation*} x^0=1 \text{ for } x \neq 0 \end{equation*}

  • Negative Exponents:

    \begin{equation*} x^{-n}=\frac{1}{x^n} \end{equation*}

  • Properties if Negative Exponents:

    \begin{equation*} \frac{1}{x^{-n}}=x^n \text{ and } \left(\frac{x}{y}\right)^{-n}=\left(\frac{y}{x}\right)^n \end{equation*}
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