Powers of 10 and Scientific Notation: How to Write Large Numbers

Science and technology use extremely large numbers (distance between planets, number of atoms) or extremely small numbers (size of atoms, electron mass). We use powers of 10 and scientific notation to write them simply.

What are powers of 10
Positive powers of 10
Negative powers of 10
What is scientific notation (standard form)
How to convert a number to scientific notation
Examples from practice
Where is this useful?
Practice exercises
Frequently Asked Questions

What are powers of 10

Powers of 10 are numbers obtained by raising 10 to various exponents. They form the basis of our decimal system and are used to simply write very large and very small numbers.

Positive powers of 10

With a positive exponent, we move the decimal point to the right:

Overview of positive powers:

$1 0^{0} = 1$ (one)
$1 0^{1} = 10$ (ten)
$1 0^{2} = 100$ (hundred)
$1 0^{3} = 1000$ (thousand)
$1 0^{4} = 10000$ (ten thousand)
$1 0^{5} = 100000$ (hundred thousand)
$1 0^{6} = 1000000$ (million)
$1 0^{9} = 1000000000$ (billion)

Examples:

$1 0^{1} = 10$
$1 0^{2} = 100$
$1 0^{6} = 1000000$ (million)

💡 Tip: Each exponent tells you how many zeros are after the one. $1 0^{4} = 10000$ → 4 zeros.

Negative powers of 10

With a negative exponent, we move the decimal point to the left:

Overview of negative powers:

$1 0^{- 1} = \frac{1}{10} = 0.1$ (tenth)
$1 0^{- 2} = \frac{1}{100} = 0.01$ (hundredth)
$1 0^{- 3} = \frac{1}{1000} = 0.001$ (thousandth)
$1 0^{- 6} = \frac{1}{1 0 ^{6}} = 0.000 001$ (millionth)

Examples:

$1 0^{- 1} = 0.1$
$1 0^{- 2} = 0.01$
$1 0^{- 3} = 0.001$

👉 More about negative exponents: Negative exponents – explanation

What is scientific notation (standard form)

Scientific notation (or standard form) is a way to write large and small numbers simply and clearly.

Definition

We write a number in the form:

a \cdot 1 0^{n}

where:

$a$ is a number from 1 to 10 (including decimals)
$n$ is an integer (positive or negative)

When do we use scientific notation?

Large numbers: distance from Earth to the Sun is $1.5 \cdot 1 0^{8}$ km
Small numbers: diameter of an atom is approximately $1 \cdot 1 0^{- 10}$ m

💡 Why the number from 1 to 10? So the number is "standardized" – it has only one digit before the decimal point. This is an international standard.

How to convert a number to scientific notation

Step 1: Move the decimal point

Move the decimal point so that you get a number between 1 and 10.

Step 2: Count the moves

Count how many places you moved the decimal point. This becomes the exponent.

If you moved right → exponent is negative
If you moved left → exponent is positive

Examples

Example 1: 3,500,000

Step 1: Move decimal: $3.5$
Step 2: We moved 6 places left
Step 3: Result: $3.5 \cdot 1 0^{6}$

Example 2: 0.000042

Step 1: Move decimal: $4.2$
Step 2: We moved 5 places right
Step 3: Result: $4.2 \cdot 1 0^{- 5}$

More examples

$5, 000 = 5 \cdot 1 0^{3}$
$730, 000 = 7.3 \cdot 1 0^{5}$
$0.001 = 1 \cdot 1 0^{- 3}$
$0.00056 = 5.6 \cdot 1 0^{- 4}$

Examples from practice

Astronomy

Distance from Earth to the Moon: $3.84 \cdot 1 0^{8}$ m
Distance from Earth to the Sun: $1.496 \cdot 1 0^{11}$ m
Number of stars in the Milky Way: approximately $2 \cdot 1 0^{11}$

Physics

Speed of light: $3 \cdot 1 0^{8}$ m/s
Planck's constant: $6.626 \cdot 1 0^{- 34}$ J·s
Electron mass: $9.109 \cdot 1 0^{- 31}$ kg

Chemistry

Avogadro's number: $6.022 \cdot 1 0^{23}$ mol $^{- 1}$
One atomic mass unit: $1.66 \cdot 1 0^{- 27}$ kg

Biology

Size of a cell: approximately $1 0^{- 5}$ m
Size of a virus: approximately $1 0^{- 7}$ m
Number of bacteria on Earth: approximately $5 \cdot 1 0^{30}$

Where is this useful?

Making large numbers manageable

Instead of writing 156,000,000,000,000, we write $1.56 \cdot 1 0^{14}$ .

Making calculations easier

Multiplying large numbers:

(3 \cdot 1 0^{8}) \cdot (2 \cdot 1 0^{5}) = 6 \cdot 1 0^{13}

Dividing large numbers:

\frac{6 \cdot 1 0 ^{12}}{2 \cdot 1 0 ^{5}} = 3 \cdot 1 0^{7}

Practice exercises

Interactive exercises

🔢 Basic powers
📊 Scientific notation – practice moving the decimal point
🧮 Calculating with powers
✏️ Simplifying expressions

Practice problems

Convert to scientific notation:

45,000
0.000 67
1,230,000

Convert from scientific notation:

$3.5 \cdot 1 0^{4}$
$2.1 \cdot 1 0^{- 3}$
$7.89 \cdot 1 0^{6}$

Frequently Asked Questions

Why do scientists use scientific notation?

Because atoms, stars, and molecules are either extremely large (billions of billions) or extremely small (billionths of a millimeter). Scientific notation makes these numbers readable and easier to work with.

Can the exponent be zero?

Yes! $1 0^{0} = 1$ , so $5 \cdot 1 0^{0} = 5$ .

What's the difference between $1 0^{3}$ and $3 \cdot 1 0^{1}$ ?

1 0^{3} = 1000

3 \cdot 1 0^{1} = 30

These are completely different values!

How do I multiply numbers in scientific notation?

Multiply the coefficients and add the exponents:

(a \cdot 1 0^{m}) \cdot (b \cdot 1 0^{n}) = (a \cdot b) \cdot 1 0^{m + n}

How do I divide numbers in scientific notation?

Divide the coefficients and subtract the exponents:

\frac{a \cdot 1 0 ^{m}}{b \cdot 1 0 ^{n}} = \frac{a}{b} \cdot 1 0^{m - n}

Summary

SI Prefixes:

Kilo- ( $1 0^{3}$ ): 1,000
Mega- ( $1 0^{6}$ ): 1,000,000
Giga- ( $1 0^{9}$ ): 1,000,000,000
Tera- ( $1 0^{12}$ ): 1,000,000,000,000
Milli- ( $1 0^{- 3}$ ): 0.001
Micro- ( $1 0^{- 6}$ ): 0.000001
Nano- ( $1 0^{- 9}$ ): 0.000000001
Pico- ( $1 0^{- 12}$ ): 0.000000000001