Lesson 5: Integer Exponents and Scientific Notation

New Concept

We'll expand exponent rules to include negative integers, where $a^{-n} = \frac{1}{a^n}$. You'll apply this to simplify expressions and use scientific notation to efficiently write and calculate with very large or small numbers.

What’s next

Next, you'll tackle interactive examples and practice problems to master negative exponents and scientific notation conversions.

Property

If $n$ is a positive integer and $a \neq 0$, then $a^{-n} = \frac{1}{a^n}$.

The negative exponent tells us to re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent. Any expression that has negative exponents is not considered to be in simplest form.

Examples

• To simplify $4^{-2}$, we take the reciprocal of the base and make the exponent positive: $4^{-2} = \frac{1}{4^2} = \frac{1}{16}$.

Property

If $a$, $b$ are real numbers and $m$, $n$ are integers, then:

Examples

• Using the Product Property: $y^{-6} \cdot y^4 = y^{-6+4} = y^{-2} = \frac{1}{y^2}$.

Property

A number is expressed in scientific notation when it is of the form

where $a \ge 1$ and $a < 10$ and $n$ is an integer. Scientific notation is a useful way of writing very large or very small numbers.

Examples

• For a large number like 4,000, we write it as $4 \times 1000$, which becomes $4 \times 10^3$ in scientific notation.

• For a small number like 0.004, we write it as $4 \times \frac{1}{1000}$, which becomes $4 \times 10^{-3}$ in scientific notation.

Property

To convert from decimal notation 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 is $10^n$. If the original number is between 0 and 1, the power is $10^{-n}$.

Examples

• To convert 37,000, move the decimal 4 places to the left to get 3.7. Since the original number was greater than 1, the result is $3.7 \times 10^4$.

• To convert 0.0052, move the decimal 3 places to the right to get 5.2. Since the original number was between 0 and 1, the result is $5.2 \times 10^{-3}$.

Property

To convert scientific notation to decimal form:

1. Determine the exponent, $n$, on the factor 10.
2. Move the decimal $n$ places. 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. Add zeros as needed.

Examples

• To convert $6.2 \times 10^3$ to decimal form, move the decimal point 3 places to the right, which gives 6,200.

• To convert $8.9 \times 10^{-2}$ to decimal form, move the decimal point 2 places to the left, which gives 0.089.

Property

To multiply and divide numbers in scientific notation, use the Properties of Exponents.

For multiplication, multiply the decimal numbers and add the exponents of the powers of 10: $(a \times 10^m)(b \times 10^n) = (a \cdot b) \times 10^{m+n}$.

For division, divide the decimal numbers and subtract the exponents: $\frac{a \times 10^m}{b \times 10^n} = (\frac{a}{b}) \times 10^{m-n}$.