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

Quick answer: To multiply numbers in scientific notation, multiply the coefficients and add the exponents of 10. To divide, divide the coefficients and subtract the exponents. Then fix the coefficient so it is at least 1 and less than 10. Example: (4 × 10³)(5 × 10⁶) = 20 × 10⁹ = 2 × 10¹⁰.

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.

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Multiply and Divide in Scientific Notation: pop-up practice

Answer these quick questions, then use the feedback to decide which part of the lesson to review.

Choose an answer to begin.

1. \(a^4\cdot a^2=\)

A\(a^6\)B\(a^8\)C\(2a^6\)

2. \((x^3)^2=\)

A\(x^6\)B\(x^5\)C\(x^9\)

3. \(b^0=\)

A\(1\)B\(0\)C\(b\)

Worksheet Maker✏Practice DrillsFlashcards

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.

Multiply, Divide, Tidy

Worked Examples

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

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

Your Turn: Multiply & Divide

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

Harder Cases: Negative Exponents and Small Coefficients

The steps do not change when exponents are negative. Just be careful with the signs when you add or subtract them.

Example 1: A negative exponent in a product

(4 × 10⁻³)(2.5 × 10⁶): multiply 4 × 2.5 = 10 and add −3 + 6 = 3. That gives 10 × 10³. Since 10 is not less than 10, rewrite it as 1 × 10¹, so the answer is 1 × 10⁴.

Example 2: A coefficient below 1

(2 × 10⁵) ÷ (8 × 10²): divide 2 ÷ 8 = 0.25 and subtract 5 − 2 = 3. That gives 0.25 × 10³. Move the decimal one place right and lower the exponent by one: 2.5 × 10².

Example 3: Dividing by a negative power

(6 × 10⁻⁴) ÷ (3 × 10⁻⁹): divide 6 ÷ 3 = 2 and subtract −4 − (−9) = 5. The answer is 2 × 10⁵.

Example 4: A real-world problem

Light travels about 3 × 10⁸ meters per second. Sunlight takes about 5 × 10² seconds to reach Earth. Distance = rate × time = (3 × 10⁸)(5 × 10²) = 15 × 10¹⁰ = 1.5 × 10¹¹ meters, which is about 150 billion meters.

A quick rule for re-normalizing: if you move the decimal left, raise the exponent; if you move it right, lower the exponent.

Frequently Asked Questions

What do I do if the coefficient ends up less than 1?

Move the decimal point to the right until the coefficient is between 1 and 10, and lower the exponent by one for each place you move. For example, 0.4 × 10⁶ becomes 4 × 10⁵. The value does not change; you are only writing it in proper form.

How do you multiply three numbers in scientific notation?

Use the same steps for all three at once. Multiply the three coefficients together and add all three exponents. For (2 × 10³)(3 × 10⁻¹)(5 × 10⁴), the coefficients give 30 and the exponents give 6, so you get 30 × 10⁶, which re-normalizes to 3 × 10⁷.

Is adding scientific notation different from multiplying?

Yes. To add or subtract, the powers of 10 must match first, so you may need to rewrite one number. Then add or subtract the coefficients and keep the power. For example, 3 × 10⁴ + 5 × 10³ = 3 × 10⁴ + 0.5 × 10⁴ = 3.5 × 10⁴. Multiplying and dividing do not need matching powers.

Can I check my answer with a calculator?

Yes. Most scientific calculators have a key labeled EE or EXP for entering powers of 10, so 3 × 10⁸ is typed as 3 EE 8. Work the problem by hand first, then compare. If the calculator shows a different power of 10, look at your re-normalizing step.

Related Topics

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