Summary: Simplifying Expressions With Negative Exponents

Key Concepts

• Summary of Exponent Properties
  • If [latex]a,b[/latex] are real numbers and [latex]m,n[/latex] are integers, then [latex]\begin{array}{cccc}\mathbf{\text{Product Property}}\hfill & & & {a}^{m}\cdot {a}^{n}={a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & & {\left({a}^{m}\right)}^{n}={a}^{m\cdot 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},a\ne 0\hfill \\ \mathbf{\text{Zero Exponent Property}}\hfill & & & {a}^{0}=1,a\ne 0\hfill \\ \mathbf{\text{Quotient to a Power Property}}\hfill & & & {\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}},b\ne 0\hfill \\ \mathbf{\text{Definition of Negative Exponent}}\hfill & & & {a}^{-n}=\frac{1}{{a}^{n}}\hfill \end{array}[/latex]
• 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, [latex]n[/latex] , that the decimal point was moved.
  3. Write the number as a product with a power of [latex]10[/latex].
     • If the original number is greater than [latex]1[/latex], the power of [latex]10[/latex] will be [latex]{10}^{n}[/latex] .
     • If the original number is between [latex]0[/latex] and [latex]1[/latex], the power of [latex]10[/latex] will be [latex]{10}^{n}[/latex] .
  4. Check.
• Convert Scientific Notation to Decimal Form: To convert scientific notation to decimal form:
  1. Determine the exponent, [latex]n[/latex] , on the factor [latex]10[/latex].
  2. Move the decimal [latex]n[/latex] places, adding zeros if needed.
     • If the exponent is positive, move the decimal point [latex]n[/latex] places to the right.
     • If the exponent is negative, move the decimal point [latex]|n|[/latex] places to the left.
  3. Check.

Glossary

negative exponent

If [latex]n[/latex] is a positive integer and [latex]a\ne 0[/latex] , then [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex] .

scientific notation

A number expressed in the form [latex]a\times {10}^{n}[/latex], where [latex]a\ge 1[/latex] and [latex]a<10[/latex], and [latex]n[/latex] is an integer.

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