How to Multiply and Divide in Scientific Notation? (+FREE Worksheet!)
Unlike adding and subtracting in scientific notation, multiplying and dividing in scientific notation does not require the exponents to match first. Instead, you operate on the coefficients and the powers of 10 separately — which makes the process fast and clean. This lesson covers both operations with full examples, two video lessons, and practice.
What Is Multiplication and Division in Scientific Notation?
When numbers are written as \(\color{blue}{a \times 10^{m}}\) and \(\color{blue}{b \times 10^{n}}\):
Multiply: \(\color{blue}{(a \times 10^{m})(b \times 10^{n}) = (a \times b) \times 10^{m + n}}\)
Divide: \(\color{blue}{(a \times 10^{m}) \div (b \times 10^{n}) = (a \div b) \times 10^{m – n}}\)
Coefficients are handled with regular arithmetic; exponents follow the multiplication and division rules for powers of 10.
How to Multiply in Scientific Notation
Step 1: Multiply the coefficients
Treat the coefficients as ordinary decimal numbers.
Step 2: Add the exponents
Apply the multiplication property of exponents: \(\color{blue}{10^{m} \times 10^{n} = 10^{m + n}}\).
- \(\color{blue}{(2.0 \times 10^{3})(3.0 \times 10^{4}) = 6.0 \times 10^{7}}\)
Step 3: Re-normalize if needed
Ensure the coefficient remains between 1 and 10; adjust the exponent accordingly.
How to Divide in Scientific Notation
Step 1: Divide the coefficients
Step 2: Subtract the exponents
\(\color{blue}{10^{m} \div 10^{n} = 10^{m – n}}\)
- \(\color{blue}{(6.0 \times 10^{8}) \div (2.0 \times 10^{3}) = 3.0 \times 10^{5}}\)
Step-by-Step Summary
- Multiply or divide the coefficients using regular arithmetic.
- Add (multiply) or subtract (divide) the exponents of 10.
- Combine the new coefficient and the new power of 10.
- Re-normalize: adjust so the coefficient is \(\color{blue}{1 \le a < 10}\).
Watch: Multiplying and Dividing in Scientific Notation (Video Lesson)
Khan Academy works through a complete multiplying and dividing example with re-normalization:
Multiplication and Division in Scientific Notation – Worked Examples
Example 1: Multiply \(\color{blue}{(2.0 \times 10^{3})(3.0 \times 10^{4})}\).
Coefficients: \(\color{blue}{2.0 \times 3.0 = 6.0}\). Exponents: \(\color{blue}{3 + 4 = 7}\).
Answer: \(\color{blue}{6.0 \times 10^{7}}\)
Example 2: Divide \(\color{blue}{(6.0 \times 10^{8}) \div (2.0 \times 10^{3})}\).
Coefficients: \(\color{blue}{6.0 \div 2.0 = 3.0}\). Exponents: \(\color{blue}{8 – 3 = 5}\).
Answer: \(\color{blue}{3.0 \times 10^{5}}\)
Example 3: Multiply \(\color{blue}{(4.5 \times 10^{5})(2.0 \times 10^{3})}\).
Coefficients: \(\color{blue}{4.5 \times 2.0 = 9.0}\). Exponents: \(\color{blue}{5 + 3 = 8}\).
Answer: \(\color{blue}{9.0 \times 10^{8}}\)
Example 4: Divide \(\color{blue}{(9.0 \times 10^{6}) \div (3.0 \times 10^{2})}\).
Coefficients: \(\color{blue}{9.0 \div 3.0 = 3.0}\). Exponents: \(\color{blue}{6 – 2 = 4}\).
Answer: \(\color{blue}{3.0 \times 10^{4}}\)
More Practice: Multiplying in Scientific Notation Video
This Khan Academy video focuses on the multiplication step and re-normalization with a second complete example:
Exercises for Multiplication and Division in Scientific Notation
- \(\color{blue}{(3.0 \times 10^{4})(2.0 \times 10^{5})}\)
- \(\color{blue}{(8.0 \times 10^{9}) \div (4.0 \times 10^{3})}\)
- \(\color{blue}{(5.0 \times 10^{-2})(4.0 \times 10^{6})}\)
- \(\color{blue}{(6.3 \times 10^{7}) \div (9.0 \times 10^{3})}\)
- \(\color{blue}{(2.5 \times 10^{3})^{2}}\)
- \(\color{blue}{(1.2 \times 10^{8}) \div (6.0 \times 10^{4})}\)
Answers
- \(\color{blue}{6.0 \times 10^{9}}\)
- \(\color{blue}{2.0 \times 10^{6}}\)
- \(\color{blue}{2.0 \times 10^{5}}\)
- \(\color{blue}{7.0 \times 10^{3}}\)
- \(\color{blue}{6.25 \times 10^{6}}\)
- \(\color{blue}{2.0 \times 10^{3}}\)
Frequently Asked Questions
Do I need to match exponents before multiplying?
No. Unlike addition and subtraction, multiplication and division work on coefficients and exponents independently. There is no need to match exponents first.
When do I need to re-normalize after multiplying?
If the product of the coefficients \(\color{blue}{\text{ is } \ge 10}\) or < 1, adjust: move the decimal left or right by one place and increase or decrease the exponent by 1 for each move.
How do I handle very small numbers in division?
The same way: divide the coefficients and subtract the exponents. A negative result for the exponent is fine — it just means the answer is a very small number. For example, \(\color{blue}{(2.0 \times 10^{2}) \div (4.0 \times 10^{5}) = 0.5 \times 10^{-3} = 5.0 \times 10^{-4}}\).
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