Mathematics » Introducing Polynomials » Integer Exponents and Scientific Notation

# Key Concepts

## 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.

## 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

### 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.