Lesson 57: Negative Exponents, Scientific Notation for Small Numbers

New Concept

This course builds your mathematical toolkit, showing how numbers, operations, and variables work together to solve problems systematically and efficiently.

What’s next

To start, we'll explore negative exponents and how they help us write very small numbers using scientific notation, with worked examples and practice problems.

Property

For any non-zero number $a$ and any integer $n$, $a^{-n} = \frac{1}{a^n}$.

Examples

$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$
$10^{-4} = \frac{1}{10^4} = \frac{1}{10000}$
$5^{-3} = \frac{1}{5^3} = \frac{1}{125}$

Explanation

A negative exponent is a secret signal to flip the number! Instead of multiplying by the base, you are dividing. This cool trick transforms a power into its reciprocal, turning it into a fraction. It’s like a shrinking ray for numbers, perfect for expressing tiny values without a bunch of decimals.

Property

If a number $a$ is not zero, then $a^0 = 1$.

Examples

$150^0 = 1$
$(-8)^0 = 1$
$(\frac{2}{7})^0 = 1$

Explanation

Think of the zero exponent as the great equalizer in math. Any non-zero number, whether it's gigantic or super small, gets zapped down to exactly 1 when raised to the power of zero. It is the ultimate reset button for bases, making it a handy shortcut for simplifying complex expressions.

Property

To convert a number from scientific notation with a negative exponent to standard form, shift the decimal point to the left by the number of places indicated by the exponent.

Examples

Write in standard form: $5.12 \times 10^{-7} = 0.000000512$
Write in scientific notation: $0.000071 = 7.1 \times 10^{-5}$
Write in standard form: $2.9 \times 10^{-4} = 0.00029$

Explanation

That little negative sign on the exponent is a command to make the number smaller. If you see $10^{-N}$, it means you need to hop the decimal point N places to the left, adding zeros as placeholders. It’s how scientists write tiny measurements, like the width of a hair, without tons of zeros.