Summary: Simplifying Expressions With Exponents

Key Concepts

• Exponential Notation

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.

Contribute!

Licenses & Attributions