Summary: Exponents and Scientific Notation

Key Equations

Rules of Exponents
For nonzero real numbers [latex]a[/latex] and [latex]b[/latex] and integers [latex]m[/latex] and [latex]n[/latex]
Product rule [latex]{a}^{m}\cdot {a}^{n}={a}^{m+n}[/latex]
Quotient rule [latex]\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex]
Power rule [latex]{\left({a}^{m}\right)}^{n}={a}^{m\cdot n}[/latex]
Zero exponent rule [latex]{a}^{0}=1[/latex]
Negative rule [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex]
Power of a product rule [latex]{\left(a\cdot b\right)}^{n}={a}^{n}\cdot {b}^{n}[/latex]
Power of a quotient rule [latex]{\left(\frac{a}{b}\right)}^{n}=\frac{{a}^{n}}{{b}^{n}}[/latex]

Key Concepts

  • Products of exponential expressions with the same base can be simplified by adding exponents.
  • Quotients of exponential expressions with the same base can be simplified by subtracting exponents.
  • Powers of exponential expressions with the same base can be simplified by multiplying exponents.
  • An expression with exponent zero is defined as 1.
  • An expression with a negative exponent is defined as a reciprocal.
  • The power of a product of factors is the same as the product of the powers of the same factors.
  • The power of a quotient of factors is the same as the quotient of the powers of the same factors.
  • The rules for exponential expressions can be combined to simplify more complicated expressions.
  • Scientific notation uses powers of 10 to simplify very large or very small numbers.
  • Scientific notation may be used to simplify calculations with very large or very small numbers.

Glossary

scientific notation a shorthand notation for writing very large or very small numbers in the form [latex]a\times {10}^{n}[/latex] where [latex]1\le |a|<10[/latex] and [latex]n[/latex] is an integer