How to Multiply and Divide in Scientific Notation? (+FREE Worksheet!)

How to Multiply and Divide in Scientific Notation? (+FREE Worksheet!)
Algebra 1

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 β€” then tidy the answer back into proper form. We’ll work both, with practice and a worksheet maker a tap away.

Tutor-style math help

Multiply and Divide in Scientific Notation: what to notice and how to work it

Exponents skill
Exponent rules are shortcuts for repeated multiplication. They work only when the bases and operations match the rule.

What to notice first

Identify the base before touching the exponent. Parentheses can change the base, especially with negative numbers and fractions.

Common student mistake

Do not add exponents unless you are multiplying powers with the same base. For \((x^3)^4\), multiply exponents instead.

Key formulas and cues

\(a^m\cdot a^n=a^{m+n}\)
\(\frac{a^m}{a^n}=a^{m-n}\)
\((a^m)^n=a^{mn}\)
\(a^0=1\text{ for }a\ne0\)

A reliable path

  1. Check the baseMake sure the repeated factor is the same.
  2. Match the operationMultiplication, division, and powers of powers use different exponent moves.
  3. Clean negativesMove negative exponents across the fraction bar and make them positive.

Worked examples

Multiply same bases

Example: \(x^3\cdot x^4\)
  1. The base is x in both powers.
  2. Multiplication means add exponents.
  3. 3 + 4 = 7.
Answer: \(x^7\)

Power of a power

Example: \((y^2)^5\)
  1. The whole power is raised to another power.
  2. Multiply the exponents.
  3. 2 times 5 is 10.
Answer: \(y^{10}\)
Open pop-up practice
Try one before moving on
Try: Simplify \(\frac{x^7}{x^3}\).
Answer: \(x^4\), assuming \(x\ne0\).
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
πŸ“„ Worksheet Maker✏️ Practice DrillsπŸ“‡ Flashcards
Illustration of students learning Multiplication and Division in Scientific Notation

Multiplying and dividing in scientific notation is the easy direction β€” 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 β€” then rewrite the answer in proper form.

The big idea

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 β€” add when multiplying, subtract when dividing.

The two procedures:

  1. Multiply: multiply coefficients, add exponents.
  2. Divide: divide coefficients, subtract exponents.
  3. Then: re-normalize so the coefficient is in \([1,10)\).

Multiply, Divide, Tidy

Multiply

Γ—coeff, +exp

\((3\times10^4)(2\times10^5)\)
\(= 6\times10^{9}\)
Divide

Γ·coeff, βˆ’exp

\(\dfrac{6\times10^8}{2\times10^3} = 3\times10^{5}\)
Re-normalize

Fix big/small coefficients

\(20\times10^9\) isn’t proper.

\(= 2\times10^{10}\)

Worked Examples

Coefficients in front, powers of ten behind β€” handle each separately, then tidy.

Example A β€” Multiply

Multiply \((3\times10^4)(2\times10^5)\).

  1. Multiply coefficients: \(3 \cdot 2 = 6\).
  2. Add exponents: \(4 + 5 = 9\).
  3. Combine: \(6\times10^9\).

Answer: \(6\times10^{9}\)

(3Γ—10⁴)(2Γ—10⁡)↓(3Β·2)Γ—10⁴⁺⁡↓6Γ—10⁹×coeff, +exp

Example B β€” Divide

Simplify \(\dfrac{6\times10^8}{2\times10^3}\).

  1. Divide coefficients: \(6 \div 2 = 3\).
  2. Subtract exponents: \(8 – 3 = 5\).
  3. Combine: \(3\times10^5\).

Answer: \(3\times10^{5}\)

6Γ—10⁸ / 2Γ—10³↓(6/2)Γ—10⁸⁻³↓3Γ—10⁡÷coeff, βˆ’exp

Example C β€” Re-normalize after multiplying

Multiply \((4\times10^3)(5\times10^6)\).

  1. \(4 \cdot 5 = 20\) and \(3 + 6 = 9\), giving \(20\times10^9\).
  2. \(20\) isn’t in \([1,10)\).
  3. Shift: \(2\times10^{10}\).

Answer: \(2\times10^{10}\)

(4Γ—10Β³)(5Γ—10⁢)↓20Γ—10⁹↓2Γ—10¹⁰re-normalize

Example D β€” Divide, then check form

Simplify \(\dfrac{9\times10^7}{3\times10^2}\).

  1. Divide coefficients: \(9 \div 3 = 3\).
  2. Subtract exponents: \(7 – 2 = 5\).
  3. Coefficient already in range: \(3\times10^5\).

Answer: \(3\times10^{5}\)

9Γ—10⁷ / 3Γ—10²↓(9/3)Γ—10⁷⁻²↓3Γ—10⁡

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 β€” 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 β€” \(10^4\cdot10^5=10^9\), 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 β€” handle coefficients and powers separately.

Your Turn: Multiply & Divide

Work each and write the answer in proper form. Reveal to check.

  1. \((2\times10^3)(4\times10^4)\)
  2. \(\dfrac{8\times10^9}{4\times10^2}\)
  3. \((5\times10^6)(3\times10^2)\)
  4. \(\dfrac{6\times10^5}{2\times10^{-3}}\)
Show answers
  1. \(\color{blue}{8\times10^{7}}\)
  2. \(\color{blue}{2\times10^{7}}\)
  3. \(\color{blue}{1.5\times10^{9}}\)
  4. \(\color{blue}{3\times10^{8}}\)
Keep practicing

Make Your Own Worksheet

Generate fresh scientific-notation problems with a full answer key β€” print or save as a PDF.

New problems every click β€” never the same sheet twice
Step-by-step answer key so you can self-check
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Frequently Asked Questions

Do the exponents need to match to multiply or divide?

No β€” 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 (\(\frac{10^8}{10^3}=10^5\)) β€” 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

Scientific Notation Add & Subtract in Scientific Notation Division Property of Exponents Multiplication Property of Exponents ← Algebra 1 Hub

Continue Your Study

Ready for the next step? Pick up right where this lesson leaves off:

Up nextMultiplication Property of ExponentsContinue with Exponents and VariablesOpen lesson β†’Up nextDivision Property of ExponentsContinue with Exponents and VariablesOpen lesson β†’Up nextZero and Negative ExponentsContinue with Exponents and VariablesOpen lesson β†’
Effortless Math Team

Effortless Math Team

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