6.2Scientific Notation & Operations
Write numbers in scientific notation, convert between standard and scientific form, and perform multiplication and division — with real-world applications in astronomy, biology, and engineering.
Why This Matters
Operating with scientific notation is essential in Physics and Chemistry, where you'll multiply and divide quantities like the speed of light or Avogadro's number. It's also a tested skill on the SAT and ACT.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How does scientific notation allow scientists and engineers to work efficiently with numbers that are astronomically large or incredibly small?
Lesson Overview
The distance from Earth to the Sun is 150,000,000,000 meters. The diameter of a hydrogen atom is 0.000000000106 meters. Writing and computing with numbers like these in standard form is impractical. Scientific notation is a compact, standardized way to express any number as a product of a coefficient (between 1 and 10) and a power of ten. It is the universal language of science, engineering, and technology — used in every field from astronomy to nanotechnology.
Parts of scientific notation: coefficient × 10^n
Place Value & Powers of Ten
Place Value Chart — Powers of Ten
| Billions | Millions | Thousands | Ones | Tenths | Thousandths | Millionths |
|---|---|---|---|---|---|---|
| 10⁹ | 10⁶ | 10³ | 10⁰ | 10⁻¹ | 10⁻³ | 10⁻⁶ |
| 1,000,000,000 | 1,000,000 | 1,000 | 1 | 0.1 | 0.001 | 0.000001 |
Each step left multiplies by 10. Each step right divides by 10.
Powers of Ten — Reference Table
| Power | Standard Form | Prefix | Real-World Example |
|---|---|---|---|
| 10¹² | 1,000,000,000,000 | Trillion | Distance light travels in ~1 month |
| 10⁹ | 1,000,000,000 | Billion | Approximate number of seconds in 32 years |
| 10⁶ | 1,000,000 | Million | Approximate distance to Moon (km) |
| 10³ | 1,000 | Thousand | Meters in 1 kilometer |
| 10⁰ | 1 | One | Base unit |
| 10⁻³ | 0.001 | Milli- | Thickness of a credit card (m) |
| 10⁻⁶ | 0.000001 | Micro- | Width of a human hair (m) |
| 10⁻⁹ | 0.000000001 | Nano- | Diameter of a DNA strand (m) |
| 10⁻¹² | 0.000000000001 | Pico- | Diameter of a proton (m) |
Decimal Movement — The Key Idea
Positive Exponent → Large Number
6.02 × 10⁶ = 6,020,000
1.5 × 10⁹ = 1,500,000,000
3.0 × 10²³ = very large!
Decimal moves RIGHT
Negative Exponent → Small Number
9.1 × 10⁻³¹ = 0.000...091
1.6 × 10⁻¹⁹ = 0.000...016
5.0 × 10⁻⁶ = 0.000005
Decimal moves LEFT
Conversion Flowchart
Multiplication & Division Rules
Scientific Notation Multiplication — Step-by-Step
Multiply Coefficients
(a × 10ᵐ)(b × 10ⁿ)
→ (a · b) × 10ᵐ⁺ⁿ
Multiply the front numbers normally
Add Exponents
10ᵐ × 10ⁿ = 10ᵐ⁺ⁿ
Product Rule of Exponents
Adjust if Needed
If |a·b| ≥ 10 or < 1
Re-normalize so coefficient is between 1 and 10
Scientific Notation Division — Step-by-Step
Divide Coefficients
(a × 10ᵐ) ÷ (b × 10ⁿ)
→ (a ÷ b) × 10ᵐ⁻ⁿ
Divide the front numbers normally
Subtract Exponents
10ᵐ ÷ 10ⁿ = 10ᵐ⁻ⁿ
Quotient Rule of Exponents
Adjust if Needed
If |a÷b| ≥ 10 or < 1
Re-normalize coefficient to [1, 10)
Worked Examples
Write 93,000,000 in scientific notation.
Identify the first non-zero digit: 9.
Place the decimal after the 9: 9.3000000
Count how many places the decimal moved: 7 places to the LEFT.
Moving left → positive exponent.
93,000,000 = 9.3 × 10⁷
Write 0.000045 in scientific notation.
Identify the first non-zero digit: 4.
Place the decimal after the 4: 4.5
Count how many places the decimal moved: 5 places to the RIGHT.
Moving right → negative exponent.
0.000045 = 4.5 × 10⁻⁵
Convert 6.02 × 10²³ to standard form. (Avogadro's Number)
The exponent is +23, so move the decimal 23 places to the RIGHT.
Start: 6.02
Move right 23 places, filling with zeros.
602,000,000,000,000,000,000,000
Multiply: (3.0 × 10⁴)(2.5 × 10⁶)
Step 1 — Multiply the coefficients: 3.0 × 2.5 = 7.5
Step 2 — Add the exponents (Product Rule): 10⁴ × 10⁶ = 10⁴⁺⁶ = 10¹⁰
Step 3 — Combine: 7.5 × 10¹⁰
Step 4 — Check: Is 7.5 between 1 and 10? Yes. ✓
Divide: (8.4 × 10⁹) ÷ (2.1 × 10³) — with re-normalization check
Step 1 — Divide the coefficients: 8.4 ÷ 2.1 = 4.0
Step 2 — Subtract the exponents (Quotient Rule): 10⁹ ÷ 10³ = 10⁹⁻³ = 10⁶
Step 3 — Combine: 4.0 × 10⁶
Step 4 — Check: Is 4.0 between 1 and 10? Yes. ✓
Guided Practice
Guided Practice Video: Scientific Notation Operations
Watch the guided practice walkthrough for scientific notation operations, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Write 4,700,000 in scientific notation.
Hint: Place the decimal after the first non-zero digit (4). Count how many places you moved left.
Write 0.00000082 in scientific notation.
Hint: Place the decimal after 8. Count how many places you moved right. Moving right → negative exponent.
Convert 2.9 × 10⁵ to standard form.
Hint: Positive exponent → move decimal RIGHT 5 places. Fill with zeros as needed.
Multiply: (2.0 × 10⁵)(4.0 × 10³)
Hint: Step 1: Multiply coefficients (2.0 × 4.0). Step 2: Add exponents (5 + 3). Check if coefficient needs adjustment.
Divide: (9.6 × 10⁸) ÷ (3.2 × 10⁵)
Hint: Step 1: Divide coefficients (9.6 ÷ 3.2). Step 2: Subtract exponents (8 − 5). Check coefficient.
Key Vocabulary
Scientific Notation
A number written as a × 10ⁿ where 1 ≤ |a| < 10 and n is an integer.
Coefficient
The number a in scientific notation; must satisfy 1 ≤ |a| < 10.
Power of Ten
An expression of the form 10ⁿ; determines the magnitude (size) of the number.
Standard Form
A number written out fully with all digits, such as 93,000,000.
Positive Exponent
Indicates a large number (≥ 1); the decimal moves right when converting.
Negative Exponent
Indicates a small number (between 0 and 1); the decimal moves left when converting.
Product Rule
When multiplying powers with the same base, add the exponents: 10ᵐ · 10ⁿ = 10ᵐ⁺ⁿ.
Quotient Rule
When dividing powers with the same base, subtract the exponents: 10ᵐ ÷ 10ⁿ = 10ᵐ⁻ⁿ.
Interactive Practice — 5 Questions
Which of the following is correctly written in scientific notation?
Write 0.000000082 in scientific notation.
Multiply: (3.0 × 10⁴)(2.0 × 10⁵)
Divide: (8.0 × 10⁶) ÷ (4.0 × 10²)
After multiplying (5.0 × 10³)(4.0 × 10⁵), a student gets 20 × 10⁸. What is the re-normalized answer?
Independent Practice
Independent Practice
Write 8,200,000 in scientific notation. Show how you counted the decimal places.
Write 0.00000047 in scientific notation. Identify whether the exponent is positive or negative and explain why.
Convert 3.5 × 10⁶ to standard form. Then convert 8.1 × 10⁻⁴ to standard form.
Multiply: (2.0 × 10³)(3.0 × 10⁴). Show all steps and verify the coefficient is in [1, 10).
Divide: (7.2 × 10⁷) ÷ (9.0 × 10⁴). Show all steps. Then write a real-world sentence that could be modeled by this calculation.
Common Mistakes
Coefficient out of range — writing 45 × 10³ and calling it scientific notation.
The coefficient must satisfy 1 ≤ |a| < 10. Correct: 4.5 × 10⁴.
Wrong sign on exponent — writing 3.2 × 10³ for the small number 0.0032.
Small numbers (between 0 and 1) need a negative exponent. Correct: 3.2 × 10⁻³.
Multiplying the exponents instead of adding them: (10³)(10⁴) = 10¹².
Product Rule: ADD the exponents. (10³)(10⁴) = 10⁷.
Forgetting to re-normalize after multiplying — leaving a result like 20 × 10⁸.
If the coefficient is ≥ 10 or < 1, adjust: 20 × 10⁸ = 2.0 × 10⁹.
Math Tips
Count every decimal place moved — including zeros between the decimal and the first non-zero digit.
Coefficient check: after converting, always verify 1 ≤ |a| < 10. If not, adjust the decimal and exponent.
Multiplication shortcut: multiply the coefficients normally, then ADD the exponents (Product Rule).
Division shortcut: divide the coefficients normally, then SUBTRACT the exponents (Quotient Rule).
Re-normalize: if your coefficient after an operation is ≥ 10 or less than 1, move the decimal and adjust the exponent accordingly.