Lesson 7: Integer Exponents and Scientific Notation

New Concept

This lesson expands exponent rules to include negative integers, like $a^{-n} = \frac{1}{a^n}$. You'll use this to simplify expressions and master scientific notation—a powerful way to write very large or small numbers.

What’s next

Next, you'll tackle interactive practice cards and worked examples that build your skills for problems involving scientific notation.

Property

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

Property of Negative Exponents
If $n$ is an integer and $a \neq 0$, then $\frac{1}{a^{-n}} = a^n$.

Quotient to a Negative Exponent Property
If $a$ and $b$ are real numbers, $a \neq 0, b \neq 0$, and $n$ is an integer, then $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$.

Property

If $a$ and $b$ are real numbers, and $m$ and $n$ are integers, then:
Product Property: $a^m \cdot a^n = a^{m+n}$
Power Property: $(a^m)^n = a^{mn}$
Product to a Power: $(ab)^m = a^m b^m$
Quotient Property: $\frac{a^m}{a^n} = a^{m-n}, a \neq 0$
Zero Exponent Property: $a^0 = 1, a \neq 0$
Quotient to a Power Property: $(\frac{a}{b})^m = \frac{a^m}{b^m}, b \neq 0$
Properties of Negative Exponents: $a^{-n} = \frac{1}{a^n}$ and $\frac{1}{a^{-n}} = a^n$
Quotient to a Negative Exponent: $(\frac{a}{b})^{-n} = (\frac{b}{a})^n$

Examples

• Using the Product Property: $x^5 \cdot x^{-2} = x^{5+(-2)} = x^3$.

• Using the Power Property: $(y^{-3})^4 = y^{-3 \cdot 4} = y^{-12} = \frac{1}{y^{12}}$.

Property

Scientific Notation
A number is expressed in scientific notation when it is of the form $a \times 10^n$ where $1 \le |a| < 10$ and $n$ is an integer.

How to Convert from 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 number is between 0 and 1, the power will be $10^{-n}$.

Examples

• To write 8,300,000 in scientific notation, move the decimal 6 places to the left to get 8.3. So, the notation is $8.3 \times 10^6$.

Property

How 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.

Examples

• Convert $4.5 \times 10^5$ to decimal form. The exponent is positive 5, so move the decimal 5 places to the right to get 450,000.

• Convert $7.1 \times 10^{-3}$ to decimal form. The exponent is negative 3, so move the decimal 3 places to the left to get 0.0071.

Property

To multiply and divide numbers in scientific notation, use the Commutative Property to rearrange the factors. Group the decimal numbers and the powers of 10 separately. Multiply or divide the decimal numbers, and then use the exponent properties to multiply or divide the powers of 10.

Examples

• Multiply $(3 \times 10^4)(2 \times 10^2)$. This is $(3 \cdot 2) \times (10^4 \cdot 10^2) = 6 \times 10^6$.

• Divide $\frac{8 \times 10^7}{4 \times 10^3}$. This is $(\frac{8}{4}) \times (\frac{10^7}{10^3}) = 2 \times 10^{7-3} = 2 \times 10^4$.