Scientific Notation Operations Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Scientific Notation Operations.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Performing addition, subtraction, multiplication, and division on numbers expressed in scientific notation.

Multiplying and dividing are straightforward: multiply or divide the coefficients and add or subtract the exponents. Adding and subtracting require matching the powers of 10 first, like finding a common denominator.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Combine the number parts with the matching operation while the powers of ten follow the exponent rules.

Common stuck point: The procedure for scientific notation operations is the easy part; the trap is adding exponents when adding the numbers. Asking "Are both numbers in $a\times 10^m$ form, and is the operation multiply/divide (combine exponents) or add/subtract (match exponents first)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Are both numbers in $a\times 10^m$ form, and is the operation multiply/divide (combine exponents) or add/subtract (match exponents first)?

Worked Examples

Example 1

medium

Compute

(3.2 \times 10^5) \times (4.0 \times 10^{-3})

and express in scientific notation.

Answer

1.28 \times 10^3

First step

Multiply the coefficients:

3.2 \times 4.0 = 12.8

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Example 2

hard

Compute

\dfrac{6.0 \times 10^8}{2.4 \times 10^{-2}}

and

(5.0 \times 10^3) + (3.0 \times 10^2)

, expressing both in scientific notation.

Example 3

medium

Multiply

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

and normalize to scientific notation.

Example 4

medium

Add

5 \times 10^4 + 3 \times 10^3

. Express in scientific notation.

Example 5

medium

Subtract

4 \times 10^5 - 2 \times 10^4

. Express in scientific notation.

Example 6

hard

A computer performs

2.5 \times 10^9

operations per second. How many operations in

4.0 \times 10^{-3}

seconds?

Example 7

hard

Light travels at

3 \times 10^8

m/s. How far does it travel in one nanosecond (

10^{-9}

s)?

Example 8

hard

Earth's mass is

\approx 6 \times 10^{24}

kg and Sun's mass is

\approx 2 \times 10^{30}

kg. How many Earth masses make up the Sun?

Example 9

challenge

A drop of water contains about

1.7 \times 10^{21}

molecules. If a swimming pool holds

2.5 \times 10^5

liters of water, and a drop is

5 \times 10^{-5}

liters, estimate the number of water molecules in the pool.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Compute

(2.5 \times 10^4) \times (3.0 \times 10^6)

. Express your answer in scientific notation.

Example 2

medium

The mass of the Earth is

5.97 \times 10^{24}

kg. The mass of the Moon is

7.35 \times 10^{22}

kg. How many times heavier is the Earth than the Moon?

Example 3

easy

Multiply:

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

Example 4

easy

Divide:

\dfrac{8 \times 10^9}{4 \times 10^2}

Example 5

easy

Multiply:

(5 \times 10^6)(1 \times 10^{-2})

Example 6

easy

Add:

3 \times 10^5 + 4 \times 10^5

Example 7

easy

Divide:

\dfrac{9 \times 10^{-3}}{3 \times 10^{-5}}

Example 8

easy

Write

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

in scientific notation.

Example 9

easy

Subtract:

8 \times 10^4 - 5 \times 10^4

Example 10

easy

Multiply:

(6 \times 10^{-4})(2 \times 10^{-1})

Example 11

medium

Add:

3 \times 10^4 + 2 \times 10^3

. Give the answer in scientific notation.

Example 12

medium

Compute

\dfrac{(6 \times 10^8)(4 \times 10^{-3})}{8 \times 10^2}

Example 13

medium

Subtract:

7.5 \times 10^6 - 5 \times 10^5

Example 14

medium

Multiply:

(5 \times 10^7)(6 \times 10^4)

and write in scientific notation.

Example 15

medium

Divide:

\dfrac{3 \times 10^4}{6 \times 10^7}

in scientific notation.

Example 16

medium

Add:

9 \times 10^3 + 9 \times 10^3

in scientific notation.

Example 17

medium

A signal travels

3 \times 10^8

m/s for

2 \times 10^{-3}

s. How far does it go?

Example 18

medium

Compute

(2 \times 10^3)^4

in scientific notation.

Example 19

medium

Subtract:

1.2 \times 10^{-2} - 4 \times 10^{-3}

Example 20

challenge

\dfrac{a \times 10^m}{2 \times 10^3} = 4 \times 10^5

, and

1 \le a < 10

, find

a

and

m

Example 21

challenge

Estimate

\dfrac{(7.0 \times 10^9) + (3.0 \times 10^9)}{5 \times 10^{-2}}

in scientific notation.

Example 22

challenge

Compute

\dfrac{(4 \times 10^{-6})^2}{8 \times 10^{-9}}

in scientific notation.

Example 23

easy

Multiply:

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

Example 24

easy

Divide:

\dfrac{6 \times 10^7}{3 \times 10^2}

Example 25

easy

Add:

4 \times 10^6 + 2 \times 10^6

Example 26

easy

Subtract:

9 \times 10^{-3} - 4 \times 10^{-3}

Example 27

easy

Multiply:

(1.5 \times 10^2)(2 \times 10^3)

Example 28

medium

Divide:

\dfrac{1.2 \times 10^9}{4 \times 10^3}

and express in scientific notation.

Example 29

medium

Compute

(2 \times 10^{-3})^3

in scientific notation.

Example 30

medium

Compute

(3 \times 10^4)^2

in scientific notation.

Example 31

medium

A virus is about

1.2 \times 10^{-7}

m wide. How many fit in a row

6 \times 10^{-3}

m long?

Example 32

medium

Multiply:

(4.5 \times 10^{-2})(6 \times 10^{8})

and normalize.

Example 33

hard

Compute

\dfrac{(6 \times 10^4)(2 \times 10^{-1})}{4 \times 10^2}

Example 34

hard

Add

7.4 \times 10^{-6} + 5.2 \times 10^{-7}

in scientific notation.

Example 35

hard

Express the result of

(2 \times 10^3)^4

in scientific notation.

Example 36

hard

Multiply:

(8 \times 10^{-4})(5 \times 10^{-6})

. Normalize.

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Related Concepts

Scientific Notation Exponent Rules Significant Figures Estimation

Background Knowledge

These ideas may be useful before you work through the harder examples.

scientific notationexponent rules