How to Multiply and Divide in Scientific Notation? (+FREE Worksheet!)
Multiplication and Division in Scientific Notation
Multiplying and dividing in scientific notation is the easy part: multiply (or divide) the coefficients and add (or subtract) the exponents — then tidy the answer back into proper form. We’ll work both, with practice and a worksheet maker a tap away.
Multiply and Divide in Scientific Notation: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Check the baseMake sure the repeated factor is the same.
- Match the operationMultiplication, division, and powers of powers use different exponent moves.
- Clean negativesMove negative exponents across the fraction bar and make them positive.
Worked examples
Multiply same bases
- The base is x in both powers.
- Multiplication means add exponents.
- 3 + 4 = 7.
Power of a power
- The whole power is raised to another power.
- Multiply the exponents.
- 2 times 5 is 10.
Try one before moving on
Multiply and Divide in Scientific Notation: pop-up practice
Multiplying and dividing in scientific notation is the easy direction — easier than adding, in fact, because the exponents don’t have to match. You handle the coefficients and the powers of ten separately, then tidy the result. If you know the multiplication and division properties of exponents, you already know the hard part.
In short: to multiply, multiply the coefficients and add the exponents; to divide, divide the coefficients and subtract the exponents — then rewrite the answer in proper form.
Coefficients and Powers, Separately
A scientific-notation number is a coefficient times a power of ten, and multiplication lets you regroup: \((3\times10^4)(2\times10^5) = (3\cdot2)\times(10^4\cdot10^5)\). The coefficients multiply normally; the powers of ten follow the exponent rules — add when multiplying, subtract when dividing.
The two procedures:
- Multiply: multiply coefficients, add exponents.
- Divide: divide coefficients, subtract exponents.
- Then: re-normalize so the coefficient is in \([1,10)\).
Multiply, Divide, Tidy
×coeff, +exp
\(= 6\times10^{9}\)
÷coeff, −exp
Fix big/small coefficients
\(20\times10^9\) isn’t proper.
Worked Examples
Coefficients in front, powers of ten behind — handle each separately, then tidy.
Example A — Multiply
Multiply \((3\times10^4)(2\times10^5)\).
- Multiply coefficients: \(3 \cdot 2 = 6\).
- Add exponents: \(4 + 5 = 9\).
- Combine: \(6\times10^9\).
Answer: \(6\times10^{9}\)
Example B — Divide
Simplify \(\dfrac{6\times10^8}{2\times10^3}\).
- Divide coefficients: \(6 \div 2 = 3\).
- Subtract exponents: \(8 – 3 = 5\).
- Combine: \(3\times10^5\).
Answer: \(3\times10^{5}\)
Example C — Re-normalize after multiplying
Multiply \((4\times10^3)(5\times10^6)\).
- \(4 \cdot 5 = 20\) and \(3 + 6 = 9\), giving \(20\times10^9\).
- \(20\) isn’t in \([1,10)\).
- Shift: \(2\times10^{10}\).
Answer: \(2\times10^{10}\)
Example D — Divide, then check form
Simplify \(\dfrac{9\times10^7}{3\times10^2}\).
- Divide coefficients: \(9 \div 3 = 3\).
- Subtract exponents: \(7 – 2 = 5\).
- Coefficient already in range: \(3\times10^5\).
Answer: \(3\times10^{5}\)
Where You’ll Use It
This is the everyday arithmetic of science. Computing how many atoms are in a sample, how far light travels in a year (\(3\times10^8\) m/s times the seconds in a year), or how a tiny mass scales up — all of it is “multiply the fronts, combine the powers of ten.” It keeps enormous and microscopic numbers manageable.
Mistakes to Avoid
- Adding exponents when dividing. Division subtracts: \(\frac{10^8}{10^3}=10^5\), not \(10^{11}\).
- Multiplying the exponents. When multiplying numbers, you add the exponents — \(10^4\cdot10^5=10^9\), not \(10^{20}\).
- Skipping re-normalization. \(20\times10^9\) must become \(2\times10^{10}\); \(0.5\times10^6\) must become \(5\times10^5\).
- Mixing up the operations. Unlike addition, the exponents do not need to match here — handle coefficients and powers separately.
Your Turn: Multiply & Divide
Work each and write the answer in proper form. Reveal to check.
- \((2\times10^3)(4\times10^4)\)
- \(\dfrac{8\times10^9}{4\times10^2}\)
- \((5\times10^6)(3\times10^2)\)
- \(\dfrac{6\times10^5}{2\times10^{-3}}\)
Show answers
- \(\color{blue}{8\times10^{7}}\)
- \(\color{blue}{2\times10^{7}}\)
- \(\color{blue}{1.5\times10^{9}}\)
- \(\color{blue}{3\times10^{8}}\)
Make Your Own Worksheet
Generate fresh scientific-notation problems with a full answer key — print or save as a PDF.
Frequently Asked Questions
Do the exponents need to match to multiply or divide?
No — that’s only for addition and subtraction. To multiply or divide, you handle the coefficients and the powers of ten separately, so the exponents can be different.
What do I do with the exponents?
Add them when multiplying (\(10^4\cdot10^5=10^9\)) and subtract them when dividing (\(\frac{10^8}{10^3}=10^5\)) — the same multiplication and division properties of exponents.
What does “re-normalize” mean?
Put the answer back into proper form with a coefficient in \([1,10)\). \(20\times10^9\) becomes \(2\times10^{10}\); \(0.5\times10^6\) becomes \(5\times10^5\).
How do I divide by a negative power, like \(10^{-3}\)?
Subtracting a negative adds: \(\frac{10^5}{10^{-3}} = 10^{5-(-3)} = 10^8\).
Related Topics
Continue Your Study
Ready for the next step? Pick up right where this lesson leaves off:
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