Mathematics » Introducing Polynomials » Integer Exponents and Scientific Notation

Key Concepts

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Key Concepts

  • Summary of Exponent Properties
    • If \(a,b\) are real numbers and \(m,n\) are integers, then
      \(\begin{array}{cccc}\mathbf{\text{Product Property}}\hfill & & & {a}^{m}·{a}^{n}={a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & & {\left({a}^{m}\right)}^{n}={a}^{m·n}\hfill \\ \mathbf{\text{Product to a Power Property}}\hfill & & & {\left(ab\right)}^{m}={a}^{m}{b}^{m}\hfill \\ \mathbf{\text{Quotient Property}}\hfill & & & \frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\phantom{\rule{0.2em}{0ex}}a\ne 0\hfill \\ \mathbf{\text{Zero Exponent Property}}\hfill & & & {a}^{0}=1,\phantom{\rule{0.2em}{0ex}}a\ne 0\hfill \\ \mathbf{\text{Quotient to a Power Property}}\hfill & & & {\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}},\phantom{\rule{0.2em}{0ex}}b\ne 0\hfill \\ \mathbf{\text{Definition of Negative Exponent}}\hfill & & & {a}^{-n}=\frac{1}{{a}^{n}}\hfill \end{array}\)

  • Convert from Decimal Notation to Scientific Notation: To convert a decimal to scientific notation:
    1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
    2. Count the number of decimal places, \(n\), that the decimal point was moved.
    3. Write the number as a product with a power of 10.
      • If the original number is greater than 1, the power of 10 will be \({10}^{n}\).
      • If the original number is between 0 and 1, the power of 10 will be \({10}^{n}\).
    4. Check.
  • Convert Scientific Notation to Decimal Form: To convert scientific notation to decimal form:
    1. Determine the exponent, \(n\), on the factor 10.
    2. Move the decimal \(n\) places, adding zeros if needed.
      • If the exponent is positive, move the decimal point \(n\) places to the right.
      • If the exponent is negative, move the decimal point \(|n|\) places to the left.
    3. Check.


negative exponent

If \(n\) is a positive integer and \(a\ne 0\), then \({a}^{-n}=\frac{1}{{a}^{n}}\).

scientific notation

A number expressed in

scientific notation

when it is of the form \(a\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{n},\) where \(a\ge 1\) and \(a<10,\) and \(n\) is an integer.

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