Powers of 10 and Scientific Notation: How to Write Large and Small 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.

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Table of Contents

• 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

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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.

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Positive powers of 10

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

Overview of positive powers:

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

Examples:

• $10^1=10$
• $10^2=100$
• $10^6=1000000$ (million)

> 💡 Tip: Each exponent tells you how many zeros are after the one. $10^4=10000$ → 4 zeros.

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Negative powers of 10

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

Overview of negative powers:

• $10^{-1}=\frac{1}{10}=0.1$ (tenth)
• $10^{-2}=\frac{1}{100}=0.01$ (hundredth)
• $10^{-3}=\frac{1}{1000}=0.001$ (thousandth)
• $10^{-6}=\frac{1}{10^6}=0.000001$ (millionth)

Examples:

• $10^{-1}=0.1$
• $10^{-2}=0.01$
• $10^{-3}=0.001$

> 👉 More about negative exponents: Negative exponents – explanation

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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:

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 $1.5\cdot 10^8$ km
• Small numbers: diameter of an atom is approximately $1\cdot 10^{-10}$ 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.

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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: $3.5\cdot 10^6$

Example 2: 0.000042

• Step 1: Move decimal: $4.2$
• Step 2: We moved 5 places right
• Step 3: Result: $4.2\cdot 10^{-5}$

More examples

• $5,000=5\cdot 10^3$
• $730,000=7.3\cdot 10^5$
• $0.001=1\cdot 10^{-3}$
• $0.00056=5.6\cdot 10^{-4}$

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Examples from practice

Astronomy

• Distance from Earth to the Moon: $3.84\cdot 10^8$ m
• Distance from Earth to the Sun: $1.496\cdot 10^{11}$ m
• Number of stars in the Milky Way: approximately $2\cdot 10^{11}$

Physics

• Speed of light: $3\cdot 10^8$ m/s
• Planck's constant: $6.626\cdot 10^{-34}$ J·s
• Electron mass: $9.109\cdot 10^{-31}$ kg

Chemistry

• Avogadro's number: $6.022\cdot 10^{23}$ mol${}^{-1}$
• One atomic mass unit: $1.66\cdot 10^{-27}$ kg

Biology

• Size of a cell: approximately $10^{-5}$ m
• Size of a virus: approximately $10^{-7}$ m
• Number of bacteria on Earth: approximately $5\cdot 10^{30}$

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Where is this useful?

Making large numbers manageable

Instead of writing 156,000,000,000,000, we write $1.56\cdot 10^{14}$.

Making calculations easier

Multiplying large numbers:

Dividing large numbers:

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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:

• $3.5\cdot 10^4$
• $2.1\cdot 10^{-3}$
• $7.89\cdot 10^6$

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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! $10^0=1$, so $5\cdot 10^0=5$.

What's the difference between $10^3$ and $3\cdot 10^1$?

$10^3=1000$

$3\cdot 10^1=30$

These are completely different values!

How do I multiply numbers in scientific notation?

Multiply the coefficients and add the exponents:

How do I divide numbers in scientific notation?

Divide the coefficients and subtract the exponents:

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Summary

SI Prefixes:

• Kilo- ($10^3$): 1,000
• Mega- ($10^6$): 1,000,000
• Giga- ($10^9$): 1,000,000,000
• Tera- ($10^{12}$): 1,000,000,000,000
• Milli- ($10^{-3}$): 0.001
• Micro- ($10^{-6}$): 0.000001
• Nano- ($10^{-9}$): 0.000000001
• Pico- ($10^{-12}$): 0.000000000001