Dividing Monomials Summary

Key Concepts

• Quotient Property for Exponents:
  • If \(a\) is a real number, \(a\ne 0\), and \(m,n\) are whole numbers, then:
    \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},m>n\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{m-n}},n>m\)
• Zero Exponent
  • If \(a\) is a non-zero number, then \({a}^{0}=1\).
• Quotient to a Power Property for Exponents:
  • If \(a\) and \(b\) are real numbers, \(b\ne 0,\) and \(m\) is a counting number, then:
    \({\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}}\)
  • To raise a fraction to a power, raise the numerator and denominator to that power.
• Summary of Exponent Properties
  • If \(a,b\) are real numbers and \(m,n\) are whole numbers, then
    \(\begin{array}{ccccc}\mathbf{\text{Product Property}}\hfill & & \hfill {a}^{m}·{a}^{n}& =\hfill & {a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & \hfill {\left({a}^{m}\right)}^{n}& =\hfill & {a}^{m·n}\hfill \\ \mathbf{\text{Product to a Power}}\hfill & & \hfill {\left(ab\right)}^{m}& =\hfill & {a}^{m}{b}^{m}\hfill \\ \mathbf{\text{Quotient Property}}\hfill & & \hfill \frac{{a}^{m}}{{b}^{m}}& =\hfill & {a}^{m-n},a\ne 0,m>n\hfill \\ & & \hfill \frac{{a}^{m}}{{a}^{n}}& =\hfill & \frac{1}{{a}^{n-m}},a\ne 0,n>m\hfill \\ \mathbf{\text{Zero Exponent Definition}}\hfill & & \hfill {a}^{o}& =\hfill & 1,a\ne 0\hfill \\ \mathbf{\text{Quotient to a Power Property}}\hfill & & \hfill {\left(\frac{a}{b}\right)}^{m}& =\hfill & \frac{{a}^{m}}{{b}^{m}},b\ne 0\hfill \end{array}\)