Summary: Simplifying Expressions With Exponents

 

Key Concepts

  • Exponential Notation

On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.
This is read [latex]a[/latex] to the [latex]{m}^{\mathrm{th}}[/latex] power.

  • Product Property of Exponents
    • If [latex]a[/latex] is a real number and [latex]m,n[/latex] are counting numbers, then
      [latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex]
    • To multiply with like bases, add the exponents.
  • Power Property for Exponents
    • If [latex]a[/latex] is a real number and [latex]m,n[/latex] are counting numbers, then
      [latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex]
  • Product to a Power Property for Exponents
    • If [latex]a[/latex] and [latex]b[/latex] are real numbers and [latex]m[/latex] is a whole number, then
      [latex]{\left(ab\right)}^{m}={a}^{m}{b}^{m}[/latex]
  • Quotient Property of Exponents
    • If [latex]a[/latex] is a real number, [latex]a\ne 0[/latex], and [latex]m,n[/latex] are whole numbers, then [latex]{\Large\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n}[/latex].
  • The Negative Rule of Exponents
    • For any nonzero real number [latex]a[/latex] and natural number [latex]n[/latex], the negative rule of exponents states that [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex].
  • Exponents of 0 or 1
    • Any number or variable raised to a power of [latex]1[/latex] is the number itself.  [latex]n^{1}=n[/latex]
    • Any non-zero number or variable raised to a power of [latex]0[/latex] is equal to [latex]1[/latex].  [latex]n^{0}=1[/latex]
    • The quantity [latex]0^{0}[/latex] is undefined.