Summary: Simplifying Expressions With Exponents

Key Concepts

• Exponential Notation

This is read $a$ to the ${m}^{\mathrm{th}}$ power.

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

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