Lesson 51: Negative Exponents and Scientific Notation for Small Numbers

New Concept

A negative exponent indicates the reciprocal of the base raised to the positive exponent; it does not make the number negative.

Law of Exponents for Negative Exponents

What’s next

This card is just the foundation. Soon, you'll see worked examples on simplifying complex expressions and converting very small numbers into scientific notation.

Property

For any nonzero number $x$ and any integer $n$:

Examples

$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$
$10^{-2} = \frac{1}{10^2} = 0.01$
$5a^{-4} = \frac{5}{a^4}$

Explanation

A negative exponent is a secret 'flip' instruction! It doesn't make the number negative; it tells the base to move to the other side of the fraction bar. The exponent then loses its negative sign. It’s the ultimate switcheroo for simplifying expressions!

Property

Use negative powers of 10 to write numbers between 0 and 1 in scientific notation.

Examples

$1.5 \times 10^{-3}$ becomes $0.0015$.
$0.00008$ becomes $8 \times 10^{-5}$.
$0.000125$ becomes $1.25 \times 10^{-4}$.

Explanation

A negative exponent on the 10 is your guide for tiny numbers! It tells you how many places to move the decimal point to the left. This makes writing super small numbers a breeze and keeps your work tidy.

Property

To multiply same-base powers, add exponents: $a^m \cdot a^n = a^{m+n}$. To divide, subtract exponents:

Examples

$10^{-2} \cdot 10^{-4} = 10^{(-2) + (-4)} = 10^{-6}$
$\frac{10^{-2}}{10^{-4}} = 10^{(-2) - (-4)} = 10^2$
$3^2 \cdot 3^{-2} = 3^{2+(-2)} = 3^0 = 1$

Explanation

The classic exponent rules still work with negative numbers! Just be extra careful with your integer math. Remember, subtracting a negative is the same as adding a positive. The rules haven't changed, just some of the players!