Teacher Guide: Multiplying in Scientific Notation

Learn to multiply numbers in scientific notation: multiply the coefficients, add the exponents and normalize the result.

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Class quiz

10 questions on Scientific Notation. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Multiply two numbers in scientific notation by multiplying coefficients and adding exponents
Normalize a product whose coefficient is 10 or more
Multiply numbers with negative exponents correctly
Raise a number in scientific notation to a power
Solve real problems (distance, counts, data storage) with products in scientific notation

Prerequisites

• Writing numbers in scientific notation (lessons Scientific Notation and Writing Numbers in Scientific Notation)
• Product rule for exponents: $a^m \times a^n = a^{m+n}$
• Adding positive and negative integers
• Multiplying decimals

Discussion Starters

1. Why is it faster to multiply $3 \times 10^5$ by $3.15 \times 10^7$ than $300{,}000$ by $31{,}500{,}000$ ?
2. If the coefficient gets $10$ times smaller, what must happen to the power of ten so that the value stays the same?
3. Why can the product of two coefficients never reach $100$ ?
4. Where in science or technology have you seen very large numbers multiplied together?

Common Misconceptions

Multiplying the exponents, as in $10^3 \times 10^5 = 10^{15}$

Remediation: Write $10^3 \times 10^5$ out as factors: $(10 \cdot 10 \cdot 10)(10 \cdot 10 \cdot 10 \cdot 10 \cdot 10)$ and count the tens. Use the Powers of Ten visual.

Forgetting to raise the coefficient in a power: $(2 \times 10^3)^2 = 2 \times 10^6$

Remediation: Write the power as a product: $(2 \times 10^3)(2 \times 10^3) = 4 \times 10^6$ . Both parts are squared.

Thinking $12 \times 10^8$ is wrong in value

Remediation: Show it equals $1.2 \times 10^9$ (both are $1{,}200{,}000{,}000$ ). The value is right, only the form is not scientific notation.

Differentiation Ideas

For Struggling Students:

• Start with products that need no normalizing, like $(2 \times 10^3)(4 \times 10^5)$
• Use a two-column table: coefficients on the left, exponents on the right
• Check each answer by writing the numbers out in full

For On-Level Students:

• Mix positive and negative exponents
• Normalize every product and explain why the exponent goes up
• Solve word problems with speed, time and counts

For Advanced Students:

• Multiply three or more factors and raise numbers to powers like $(5 \times 10^3)^3$
• Estimate products by rounding coefficients, then compare with the exact answer
• Compute the mass of a sample from the number of molecules and the mass of one molecule

Standards Alignment

8.EE.A.4 (CCSS.MATH.CONTENT.8.EE.A.4)
Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used
8.EE.A.1 (CCSS.MATH.CONTENT.8.EE.A.1)
Know and apply the properties of integer exponents to generate equivalent numerical expressions
8.EE.A.3 (CCSS.MATH.CONTENT.8.EE.A.3)
Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities

Lesson Resources

visualScientific Notation Multiplier
Interactive tool that groups coefficients and powers of ten, adds exponents and normalizes the result step by step
visualPowers of Ten as Factors
Shows $10^3 \times 10^5$ as three tens and five tens to explain why the exponents add
activityCosmic Calculations
Students compute a light-year, the distance to the Sun and other space distances from speed and time
worksheetNormalize It!
Products like $12 \times 10^8$ and $14 \times 10^{-8}$ to rewrite correctly

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

To multiply two numbers in scientific notation, you work with the two parts separately: the coefficients and the powers of ten.

(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}

The rule has three moves:

1.Multiply the coefficients: $a \times b$
2.Add the exponents (product rule $10^m \times 10^n = 10^{m+n}$ )
3.Normalize: if $a \times b$ is $10$ or more, move the decimal point one place to the left and add $1$ to the exponent

Example:

(4 \times 10^3)(3 \times 10^5) = 12 \times 10^8 = 1.2 \times 10^9

Because both coefficients are between

1

and

10

, their product is always between

1

and

100

. So you never have to move the decimal point more than one place.

Worked Examples

Multiply

(2 \times 10^3) \times (4 \times 10^5)

Group the coefficients and the powers of ten

(2 \times 4) \times (10^3 \times 10^5)

→ Two separate products

Multiply the coefficients

2 \times 4 = 8

Add the exponents

10^3 \times 10^5 = 10^{3+5} = 10^8

Check the coefficient

1 \le 8 < 10

, so nothing to fix →

8 \times 10^8

Answer: $(2 \times 10^3)(4 \times 10^5) = 8 \times 10^8$

Common Mistakes

Multiplying the exponents: $10^3 \times 10^5 = 10^{15}$

Why it's wrong: When you multiply powers with the same base, you count the factors of $10$ : three tens and five tens make eight tens, not fifteen.

Correct: Add the exponents: $10^3 \times 10^5 = 10^{3+5} = 10^8$ . You multiply exponents only for a power of a power, like $(10^3)^2 = 10^6$ .

Stopping at $12 \times 10^8$

Why it's wrong: The value is right, but $12$ is not between $1$ and $10$ , so the answer is not in scientific notation yet.

Correct: Normalize: $12 \times 10^8 = 1.2 \times 10^9$ .

Normalizing in the wrong direction: $12 \times 10^8 = 1.2 \times 10^7$

Why it's wrong: Making the coefficient $10$ times smaller must be balanced by making the power of ten $10$ times bigger, otherwise the value changes.

Correct: Coefficient smaller, exponent bigger: $12 \times 10^8 = 1.2 \times 10^9$ .

Losing the minus sign: $10^{-4} \times 10^7 = 10^{-11}$ or $10^{11}$

Why it's wrong: The exponents are signed numbers, and adding them follows the rules for integers.

Correct: $-4 + 7 = 3$ , so $10^{-4} \times 10^7 = 10^3$ . Write the addition out with brackets when signs mix: $(-4) + 7$ .

Why It Matters

Science is full of multiplications with huge or tiny numbers, and scientific notation turns them into quick mental math:

Astronomy: light travels $3 \times 10^5$ km every second. Multiply by the seconds in a year and you get the length of a light-year
Biology: a sample has $4 \times 10^6$ bacteria in each milliliter. How many are in a whole flask?
Computing: $4 \times 10^4$ photos of $3 \times 10^6$ bytes each fill $1.2 \times 10^{11}$ bytes of memory
Chemistry: number of molecules times the mass of one molecule gives the mass of a sample

Writing these numbers out in full means counting dozens of zeros, and one missing zero makes the answer ten times too small. With scientific notation you multiply two small numbers and add two exponents.

Real World Applications

Astronomy: How Long Is a Light-Year?

A light-year is the distance light travels in one year. Astronomers get it by multiplying the speed of light by the time, both written in scientific notation.

Example:

In $500 = 5 \times 10^2$ seconds light travels $(3 \times 10^5)(5 \times 10^2) = 15 \times 10^7 = 1.5 \times 10^8$ km, about the distance from the Sun to the Earth.

1Try It Yourself

Light travels $3 \times 10^5$ km per second. One year has about $3.15 \times 10^7$ seconds.

How many kilometers does light travel in one year? (distance = speed × time)

Step 1: Write the mathematical expression

Multiply the speed by the time:

Biology: Counting Bacteria

Microbiologists count the bacteria in a small sample and multiply by the volume to estimate the total. The numbers grow so fast that scientific notation is the only practical way to write them.

Example:

A drop of $5 \times 10^{-2}$ mL from a culture with $2 \times 10^6$ bacteria per mL holds $(2 \times 10^6)(5 \times 10^{-2}) = 10 \times 10^4 = 1 \times 10^5$ bacteria.

2Try It Yourself

A culture contains $4 \times 10^6$ bacteria in each milliliter. The flask holds $5 \times 10^2$ mL.

How many bacteria are in the whole flask?

Step 1: Write the mathematical expression

Multiply the bacteria per mL by the number of mL:

Computing: Filling a Memory Card

File sizes and storage are measured in bytes, and the numbers quickly reach billions. Engineers multiply file size by the number of files to plan how much storage a device needs.

Example:

$2 \times 10^3$ songs of $4 \times 10^6$ bytes each take $8 \times 10^9$ bytes, that is 8 gigabytes.

3Try It Yourself

One photo takes about $3 \times 10^6$ bytes. A phone stores $4 \times 10^4$ photos.

How many bytes do the photos take in total?

Step 1: Write the mathematical expression

Multiply the size of one photo by the number of photos:

Key Takeaways

1To multiply in scientific notation, multiply the coefficients and multiply the powers of ten separately
2Powers of ten are multiplied by adding the exponents: $10^m \times 10^n = 10^{m+n}$
3If the new coefficient is $10$ or more, normalize: move the decimal point one place left and add $1$ to the exponent
4With negative exponents, add the signed numbers carefully: $10^{-4} \times 10^7 = 10^3$
5A power works on both parts: $(a \times 10^m)^k = a^k \times 10^{m \times k}$
6Check your answer: the coefficient must satisfy $1 \le a < 10$

Frequently Asked Questions

Why do we add the exponents instead of multiplying them?

An exponent counts how many tens are multiplied.

10^3 \times 10^5

is three tens times five tens, which is

3 + 5 = 8

tens in a row:

10^8

Could I just write the numbers out and multiply?

Yes, the value is the same. But with numbers like

3 \times 10^5 \times 3.15 \times 10^7

you would multiply

300{,}000

31{,}500{,}000

and count many zeros. In scientific notation you multiply

3 \times 3.15

and add

5 + 7

Can the product of the coefficients be 100 or more?

Not when you multiply two numbers in scientific notation: both coefficients are less than

10

, so the product is less than

10 \times 10 = 100

. You move the decimal point at most one place. With powers or with three or more factors it can be bigger, so always check.

My calculator shows 1.2E9. What does that mean?

Calculators write scientific notation with an E:

1.2\text{E}9

means

1.2 \times 10^9

. The number after the E is the exponent.

Glossary

Scientific notation: A number written as $a \times 10^n$ with $1 \le a < 10$ and $n$ an integer
Coefficient: The number $a$ in front of the power of ten, between $1$ and $10$
Exponent: The integer $n$ in $10^n$ ; it tells how many places the decimal point moves
Product rule: $a^m \times a^n = a^{m+n}$ : multiplying powers with the same base adds the exponents
Normalize: Rewrite a result like $12 \times 10^8$ so that the coefficient is between $1$ and $10$ : $1.2 \times 10^9$

Formula Card

Multiplying in scientific notation

(a \times 10^m)(b \times 10^n) = (a \times b) \times 10^{m+n}

Multiply the coefficients, add the exponents.

Normalizing

12 \times 10^8 = 1.2 \times 10^9

Coefficient divided by $10$, exponent plus $1$. The value does not change.

Power of a number in scientific notation

(a \times 10^m)^k = a^k \times 10^{m \times k}

Raise the coefficient to the power, multiply the exponent by $k$, then normalize.

Teacher Guide: Multiplying in Scientific Notation

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

Why It Matters

Real World Applications

Astronomy: How Long Is a Light-Year?

Biology: Counting Bacteria

Computing: Filling a Memory Card

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Multiplying in scientific notation

Normalizing

Power of a number in scientific notation

More in This Topic

Scientific Notation

Writing Numbers in Scientific Notation

Converting from Scientific Notation

Comparing Numbers in Scientific Notation