Powers of 10 — Definition, Formula & Examples

Powers of 10 are numbers you get by multiplying 10 by itself a certain number of times. For example, 10² = 100 (ten multiplied by itself twice), and each power of 10 is ten times greater than the one before it.

A power of 10 is any number of the form $10^n$ , where $n$ is an integer. When $n$ is positive, the result equals 1 followed by $n$ zeros. When $n$ is zero, $10^0 = 1$ . When $n$ is negative, the result is a decimal fraction equal to $\frac{1}{10^{|n|}}$ .

Key Formula

10^n = \underbrace{10 \times 10 \times \cdots \times 10}_{n \text{ factors}}

Where:

$n$ = The exponent — how many times 10 is multiplied by itself

How It Works

Each time you multiply a number by 10, every digit shifts one place to the left, making the number ten times larger. Each time you divide by 10, every digit shifts one place to the right, making it ten times smaller. This pattern connects directly to our place value system: the ones place is

10^0

, the tens place is

10^1

, the hundreds place is

10^2

, and so on. For decimals, the tenths place is

10^{-1}

and the hundredths place is

10^{-2}

Worked Example

Problem: Multiply 3.4 by 10³.

Step 1: Find the value of the power of 10.

10^3 = 10 \times 10 \times 10 = 1{,}000

Step 2: Multiplying by 1,000 shifts each digit three places to the left.

3.4 \times 1{,}000 = 3{,}400

Answer: 3.4 × 10³ = 3,400

Visualization

Why It Matters

Powers of 10 are the backbone of our number system. Understanding them helps you read large numbers like populations or distances in science, convert between metric units (kilometers to meters, grams to milligrams), and work with scientific notation in later math courses.

Common Mistakes

Mistake: Thinking the exponent tells you the number of digits in the result rather than the number of zeros after the 1.

Correction:

10^3 = 1{,}000

, which has 4 digits total but only 3 zeros. The exponent counts the zeros, not the total digits.

Related Terms

Place Value Chart — Each column represents a power of 10
Decimal Place Value — Uses negative powers of 10
Expanded Form (Expanded Notation) — Writes numbers as sums of powers of 10
Decimal Point — Separates whole-number and fractional powers of 10
Digit — Each digit's value depends on its power-of-10 position