Multiplying and Dividing in Scientific Notation
Simple Explanation
To multiply or divide numbers in scientific notation, multiply/divide the coefficients and use the exponent rules on the powers of 10 separately, then adjust the result back into proper scientific notation if the coefficient falls outside 1 ≤ a < 10.
Why Do We Need It?
Scientific fields (astronomy, chemistry, engineering) constantly multiply and divide measurements written in scientific notation — doing this efficiently avoids expanding huge or tiny numbers by hand.
Formula
Scientific Notation
a × 10ⁿ, where 1 ≤ a < 10 and n is an integer
A compact way to write very large or very small numbers as a decimal between 1 and 10, multiplied by a power of 10.
- a
- — the coefficient — a decimal number with 1 ≤ a < 10
- n
- — the exponent (power of 10) — positive for large numbers, negative for small numbers
When to use it: Whenever a number has many digits or many leading/trailing zeros, and you need a compact, easy-to-compare form.
Worked Example
Multiply two numbers in scientific notation
Simplify (3 × 10⁴) × (2 × 10³).
Why Does This Work?
Multiplication can be regrouped freely, so (a × 10ᵐ)(b × 10ⁿ) = (a × b) × (10ᵐ × 10ⁿ), and the exponent rules from Chapter 2 handle the powers of 10 part exactly as they would for any other base.
Real-Life Example
Computing total data storage
A data centre has 5 × 10³ servers, each storing 2 × 10¹² bytes.
Total storage is (5 × 10³)(2 × 10¹²) = 10 × 10¹⁵ = 1 × 10¹⁶ bytes — multiplying in scientific notation avoids writing out 16-digit numbers by hand.
Practice
Simplify (8 × 10⁵) ÷ (4 × 10²).
MediumCommon mistake
Multiplying or dividing the exponents themselves instead of the powers of 10 they belong to — always apply the ordinary exponent rules (add for multiply, subtract for divide) to the powers of 10, and combine the coefficients separately.
Quick Review
- Multiply/divide the coefficients and the powers of 10 separately.
- Use the exponent product/quotient rules on the powers of 10.
- Re-adjust into 1 ≤ a < 10 form if the resulting coefficient falls outside that range.