Scientific Notation
Grade 8 · Math · Study Guide
Write and compute with very large and very small numbers using powers of ten.
Scientific notation is a shorthand way to write numbers that are extremely large or extremely small. A number in scientific notation looks like a × 10ⁿ, where a is a number between 1 and 10 (called the coefficient) and n is an integer exponent. For example, 93,000,000 miles (the distance from Earth to the Sun) becomes 9.3 × 10⁷.
To convert a standard number to scientific notation, move the decimal point until you have a number between 1 and 10. Count the moves — that's your exponent. Moving left gives a positive exponent (large numbers); moving right gives a negative exponent (small numbers like 0.000045 = 4.5 × 10⁻⁵).
When multiplying in scientific notation, multiply the coefficients and add the exponents. When dividing, divide the coefficients and subtract the exponents. Always check that your final answer has a coefficient between 1 and 10 — adjust if needed!
| Standard Form | Scientific Notation | Exponent Direction |
|---|---|---|
| 6,200,000 | 6.2 × 10⁶ | Decimal moved 6 left → positive |
| 0.000047 | 4.7 × 10⁻⁵ | Decimal moved 5 right → negative |
| 300 | 3.0 × 10² | Decimal moved 2 left → positive |
| 0.01 | 1.0 × 10⁻² | Decimal moved 2 right → negative |
Key terms
- Scientific Notation
- A way to express numbers as a × 10ⁿ where 1 ≤ a < 10 and n is an integer.
- Coefficient
- The number a in scientific notation; must be at least 1 and less than 10.
- Exponent
- The power n on the base 10; positive for large numbers, negative for small.
- Standard Form
- The usual way to write a number, e.g., 45,000 or 0.0023.
- Order of Magnitude
- The power of 10 closest to a number; used to compare sizes.
Flashcards
- Scientific notation form
- a × 10ⁿ where 1 ≤ a < 10 and n is a whole number exponent.
- Positive exponent in sci. notation
- Indicates a large number; decimal moved left during conversion.
- Negative exponent in sci. notation
- Indicates a small number (less than 1); decimal moved right during conversion.
- Multiplying in scientific notation
- Multiply coefficients, ADD exponents: (a × 10ᵐ)(b × 10ⁿ) = (ab) × 10^(m+n).
- Dividing in scientific notation
- Divide coefficients, SUBTRACT exponents: (a × 10ᵐ) ÷ (b × 10ⁿ) = (a/b) × 10^(m−n).
- Order of magnitude
- The power of 10 in a number's scientific notation; used to compare very different sizes.
Practice questions
Which is the correct scientific notation for 0.000082?
- 8.2 × 10⁴
- 8.2 × 10⁻⁵
- 0.82 × 10⁻⁴
- 82 × 10⁻⁶
Show answer
Answer: 8.2 × 10⁻⁵
Move decimal 5 places right to get 8.2; moving right → negative exponent.
In scientific notation, the coefficient must be greater than or equal to 1 and less than 10.
Show answer
Answer: True
This is the definition of proper scientific notation form — the coefficient must be in [1, 10).
To multiply numbers in scientific notation, multiply the coefficients and ___ the exponents.
Show answer
Answer: add
By the product of powers rule: 10ᵐ × 10ⁿ = 10^(m+n), so exponents are added.
What is (3 × 10⁴) × (2 × 10³)?
- 6 × 10⁷
- 6 × 10¹²
- 5 × 10⁷
- 6 × 10⁶
Show answer
Answer: 6 × 10⁷
3 × 2 = 6; 10⁴ × 10³ = 10⁷. Result: 6 × 10⁷.
Which number is largest?
- 4.1 × 10⁵
- 3.9 × 10⁶
- 8.2 × 10⁵
- 7.0 × 10⁴
Show answer
Answer: 3.9 × 10⁶
3.9 × 10⁶ = 3,900,000 which is larger than 4.1 × 10⁵ = 410,000.
Fun facts
- The number of stars in the observable universe is estimated at 2 × 10²³ — that's 200 sextillion stars!
- A single strand of DNA is about 2.5 × 10⁻⁹ meters wide — thinner than any visible light wavelength.
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