Summary: Simplifying Expressions With Negative Exponents

Key Concepts

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

Glossary

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 the form $a\times {10}^{n}$, where $a\ge 1$ and $a<10$, and $n$ is an integer.

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