0.6Scientific Notation
Write very large and very small numbers compactly using powers of 10. Convert between standard and scientific notation and multiply or divide numbers in scientific notation.
Why This Matters
Scientific notation is the standard language of science — astronomers use it to express distances between galaxies, and chemists use it to describe the size of atoms. You'll use it in Physics, Chemistry, and Biology throughout high school.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How does scientific notation make it practical to work with extremely large or extremely small numbers in science and engineering?
Lesson Overview
Scientific notation is a way to express very large or very small numbers using a coefficient and a power of 10. A number is in scientific notation when it is written as a × 10ⁿ, where 1 ≤ |a| < 10 and n is an integer. For large numbers, n is positive (move decimal left). For small numbers, n is negative (move decimal right). Scientific notation is used in science, astronomy, and engineering to handle numbers like the distance to stars (9.46 × 10¹⁵ m) or the size of an atom (1.2 × 10⁻¹⁰ m).
Worked Examples
Write 4,700,000 in scientific notation.
Place decimal after first nonzero digit: 4.7
Count places moved: 6 places to the left → exponent is +6
Write 0.000052 in scientific notation.
Place decimal after first nonzero digit: 5.2
Count places moved: 5 places to the right → exponent is −5
Convert 3.08 × 10⁴ to standard form.
Positive exponent → move decimal right 4 places.
3.08 → 30800
Multiply: (2.5 × 10³) × (4 × 10²)
Multiply coefficients: 2.5 × 4 = 10
Add exponents: 10³ × 10² = 10⁵
10 × 10⁵ = 1 × 10⁶ (adjust coefficient)
Divide: (9 × 10⁸) ÷ (3 × 10³)
Divide coefficients: 9 ÷ 3 = 3
Subtract exponents: 10⁸ ÷ 10³ = 10⁵
Guided Practice
Answers are in the Answer Key section.
Guided Practice Video: Scientific Notation
Watch the guided practice walkthrough for scientific notation, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Write 93,000,000 in scientific notation.
Hint: Place the decimal after 9. Count how many places you moved it.
Write 0.0000007 in scientific notation.
Hint: Move the decimal right until you have a number between 1 and 10. Count the moves.
Convert 6.02 × 10²³ to standard form.
Hint: Positive exponent — move decimal right 23 places.
Multiply: (3 × 10⁴) × (2 × 10⁵)
Hint: Multiply the coefficients, then add the exponents.
Divide: (8.4 × 10⁶) ÷ (2.1 × 10²)
Hint: Divide the coefficients, then subtract the exponents.
Key Vocabulary
Scientific Notation
A number written as a × 10ⁿ where 1 ≤ |a| < 10 and n is an integer.
Example: 4,700,000 = 4.7 × 10⁶
Standard Form
The ordinary way of writing a number.
Example: 4,500,000 or 0.000045
Coefficient
The number a in a × 10ⁿ. Must satisfy 1 ≤ |a| < 10.
Example: In 3.7 × 10⁴, the coefficient is 3.7.
Power of 10
10 raised to an integer exponent. Positive exponents make numbers larger; negative exponents make numbers smaller.
Example: 10³ = 1,000; 10⁻³ = 0.001
Positive Exponent
In scientific notation, a positive exponent means the number is ≥ 10 (large number).
Example: 5.2 × 10⁷ = 52,000,000
Negative Exponent
In scientific notation, a negative exponent means the number is between 0 and 1 (small number).
Example: 3.1 × 10⁻⁴ = 0.00031
Practice Questions
Interactive Practice — 5 Questions
Write 4,700,000 in scientific notation.
Write 0.000052 in scientific notation.
Convert 3.08 × 10⁴ to standard form.
Multiply: (2.5 × 10³) × (4 × 10²)
Divide: (9 × 10⁸) ÷ (3 × 10³)
Independent Practice
Answers are in the Answer Key section.
Independent Practice
Write 56,000 in scientific notation.
Write 0.00031 in scientific notation.
Convert 2.7 × 10⁵ to standard form.
Multiply: (3 × 10³) × (2 × 10⁴)
Divide: (6 × 10⁷) ÷ (2 × 10³)
Common Mistakes
Writing the coefficient as a number ≥ 10 — e.g., 45 × 10³ instead of 4.5 × 10⁴.
The coefficient must be at least 1 and less than 10. Adjust the exponent accordingly.
Moving the decimal the wrong direction for small numbers — e.g., writing 0.003 as 3 × 10³.
For numbers less than 1, the exponent is negative: 0.003 = 3 × 10⁻³.
Forgetting to adjust the exponent when rewriting the coefficient — e.g., changing 2.5 × 10⁴ to 25 × 10⁴.
If you move the decimal right (making the coefficient larger), decrease the exponent by the same amount.
Confusing the exponent with the number of zeros — e.g., thinking 10³ has 3 zeros after the 1.
10³ = 1,000 — three zeros, yes — but 10⁻³ = 0.001, which has zeros before the 1.
Math Tips
Memory trick: large number → move decimal LEFT → positive exponent. Small number → move decimal RIGHT → negative exponent. "Left = Large = Positive."
After multiplying or dividing in scientific notation, always check that the coefficient is between 1 and 10. If not, adjust: 15 × 10⁴ = 1.5 × 10⁵.
Calculator tip: use the EE or EXP button (not ×10^) to enter scientific notation. For 3.7 × 10⁴, press 3.7 EE 4. This avoids order-of-operations errors.
SAT/ACT tip: scientific notation questions often ask you to multiply or divide two numbers. Practice the "multiply coefficients, add/subtract exponents" method until it is automatic.