Scientific Notation Practice Problems- Worked Examples
What You Need to Know Before Starting
Scientific notation exists because writing 0.0000000000000000000000347 gets old fast. It's a way to express extremely large or small numbers without losing your sanity counting zeros. If you can't work with scientific notation comfortably, you'll struggle in chemistry, physics, and any science class that deals with real data.
This guide gives you worked examples that walk through the process step by step. Copy the method. Practice until it clicks.
The Basic Format
Every scientific notation number follows this pattern:
m × 10n
Where:
- m = the coefficient (a number between 1 and 10)
- n = the exponent (positive for large numbers, negative for small ones)
That's it. Nothing fancy. The coefficient must be at least 1 but less than 10. If your coefficient is 10 or higher, you need to adjust.
Worked Example 1: Converting a Large Number
Problem: Write 4,500,000 in scientific notation.
Step 1: Place the decimal after the first non-zero digit.
4,500,000 becomes 4.500000
Step 2: Count how many places you moved the decimal.
From 4,500,000 to 4.5, that's 6 places.
Step 3: Write it out.
4,500,000 = 4.5 × 106
Check it: 4.5 × 1,000,000 = 4,500,000. Correct.
Worked Example 2: Converting a Very Small Number
Problem: Write 0.00032 in scientific notation.
Step 1: Move the decimal to sit after the first non-zero digit.
0.00032 becomes 3.2
Step 2: Count the places moved.
From 0.00032 to 3.2, you moved the decimal 4 places to the right.
Step 3: Because you moved right (making the number bigger), the exponent is negative.
0.00032 = 3.2 × 10-4
Verification: 3.2 × 0.0001 = 0.00032. Checks out.
Worked Example 3: Multiplying Scientific Notation
Problem: Multiply (3 × 104) × (2 × 103)
Step 1: Multiply the coefficients.
3 × 2 = 6
Step 2: Add the exponents.
4 + 3 = 7
Step 3: Combine.
(3 × 104) × (2 × 103) = 6 × 107
If your coefficient ends up 10 or larger, divide by 10 and increase the exponent by 1.
Worked Example 4: Dividing Scientific Notation
Problem: Divide (8 × 106) ÷ (2 × 102)
Step 1: Divide the coefficients.
8 ÷ 2 = 4
Step 2: Subtract the exponents.
6 - 2 = 4
Step 3: Write the result.
(8 × 106) ÷ (2 × 102) = 4 × 104
Worked Example 5: Adding and Subtracting
This is where people mess up. You cannot just add or subtract the coefficients unless the exponents match.
Problem: Add (3 × 104) + (2 × 103)
Step 1: Make the exponents match. Convert 2 × 103 to the same power of 10 as the first number.
2 × 103 = 0.2 × 104
Step 2: Now add the coefficients.
3 + 0.2 = 3.2
Step 3: Keep the common exponent.
(3 × 104) + (2 × 103) = 3.2 × 104
Quick shortcut: rewrite the smaller number with the same exponent as the larger one, then add.
Quick Reference: Exponent Rules
| Operation | Rule | Example |
|---|---|---|
| Multiplication | Add exponents | 102 × 103 = 105 |
| Division | Subtract exponents | 105 ÷ 102 = 103 |
| Power to power | Multiply exponents | (102)3 = 106 |
| Negative exponent | Flip to denominator | 10-2 = 1/102 |
Common Mistakes to Watch For
- Forgetting the sign on negative exponents. Small numbers need negative exponents. Always double-check.
- Keeping the coefficient too large. If it's 10 or above, you haven't finished. Move the decimal and bump the exponent.
- Adding coefficients with different exponents. This is wrong. Match the exponents first.
- Counting decimal places incorrectly. Count slowly. One mistake here and your whole answer is wrong.
Practice Problems (Answers Below)
Try these before checking the solutions:
- Write 0.00000789 in scientific notation
- Write 5.2 × 10-3 as a decimal
- Multiply (4 × 105) × (3 × 102)
- Divide (9 × 108) ÷ (3 × 104)
- Add (5 × 103) + (4.2 × 102)
Answers
- 7.89 × 10-6
- 0.0052
- 12 × 107 = 1.2 × 108 (adjusted)
- 3 × 104
- 5.42 × 103
When You'll Actually Use This
Chemistry uses it for Avogadro's number (6.02 × 1023). Physics uses it for the speed of light (3 × 108 m/s). Biology uses it for cell sizes. Engineering uses it for tolerances. Astronomy uses it for distances between stars.
You can't escape it. Master it now or keep struggling with it forever.
How to Practice Effectively
- Convert numbers you see in real life — receipts, measurements, statistics
- Set a timer and do 10 conversions daily until it's automatic
- Always verify by converting back to decimal form
- Use the exponent rules table above until you have them memorized
The goal is to do this without thinking. When it becomes automatic, you'll wonder why it ever seemed hard.