Lesson 2: Negative Exponents and Scientific Notation

New Concept

This lesson extends exponent rules to include zero and negative values, defining $a^{-n}$ as $\frac{1}{a^n}$. We'll use this to master scientific notation, a powerful method for writing and computing with very large or small numbers.

What’s next

Next, you'll tackle interactive examples and practice cards to master simplifying expressions and converting numbers into scientific notation.

Property

$a^0 = 1$, if $a \neq 0$

This definition is based on the second law of exponents. For any non-zero number $a$, the quotient $\frac{a^n}{a^n}$ is equal to 1. Using the law of exponents, we can also write $\frac{a^n}{a^n} = a^{n-n} = a^0$. Therefore, it is logical to define $a^0$ as 1.

Examples

• For a positive integer, $8^0 = 1$.
• For a negative integer, $(-55)^0 = 1$.
• For an algebraic term where variables are non-zero, $(2ab^2)^0 = 1$.

Property

$a^{-n} = \frac{1}{a^n}$ if $a \neq 0$

A negative exponent indicates the reciprocal of the power with the positive exponent. This means a factor can be moved from the numerator to the denominator (or vice versa) of a fraction by changing the sign of its exponent. For example, $\frac{1}{a^{-n}} = a^n$.

Examples

• To write without a negative exponent: $5^{-3} = \frac{1}{5^3} = \frac{1}{125}$.
• For a fraction raised to a negative power, take the reciprocal of the fraction and make the exponent positive: $(\frac{2}{3})^{-4} = (\frac{3}{2})^4 = \frac{81}{16}$.
• To rewrite a fraction using a negative exponent: $\frac{5}{y^2} = 5y^{-2}$.

Property

The laws of exponents apply to all integer exponents, including positive, negative, and zero.

1. $a^m \cdot a^n = a^{m+n}$
2. $\frac{a^m}{a^n} = a^{m-n}$ $(a \neq 0)$
3. $(a^m)^n = a^{mn}$
4. $(ab)^n = a^n b^n$
5. $(\frac{a}{b})^n = \frac{a^n}{b^n}$ $(b \neq 0)$

Examples

• Using Law 1 (Product of Powers): $y^3 \cdot y^{-7} = y^{3+(-7)} = y^{-4} = \frac{1}{y^4}$.
• Using Law 2 (Quotient of Powers): $\frac{x^4}{x^{-2}} = x^{4 - (-2)} = x^{4+2} = x^6$.
• Using Laws 3 and 4 (Power of a Power/Product): $(2x^{-4})^3 = 2^3(x^{-4})^3 = 8x^{-12} = \frac{8}{x^{12}}$.

Explanation

These five laws are your complete toolkit for simplifying expressions involving exponents. They provide rules for what to do when you multiply, divide, raise a power to another power, or distribute a power over a product or quotient.

Property

A number is in scientific notation if it is expressed as the product of a number between 1 and 10 and a power of 10.

To Write a Number in Scientific Notation.

1. Locate the decimal point so that there is exactly one nonzero digit to its left.
2. Count the number of places you moved the decimal point: this determines the power of 10.

a. If the original number is greater than 10, the exponent is positive.
b. If the original number is less than 1, the exponent is negative.

Examples

• To write a large number in scientific notation: $475,000,000 = 4.75 \times 10^8$.
• To write a small number in scientific notation: $0.000082 = 8.2 \times 10^{-5}$.
• To perform a calculation: $\frac{9.6 \times 10^8}{3 \times 10^3} = (\frac{9.6}{3}) \times 10^{8-3} = 3.2 \times 10^5$.