Scientific notation rules addition subtraction multiplication division

What does it mean?

Definitions:

Scientific Notation is the expression of a number n in the form $$a * 10^ b$$ where a is an integer such that $$1≤ |a| <10$$ and b is an integer too.

Multiplication: To multiply numbers in scientific notation, multiply the decimal numbers. Then add the exponents of the powers of 10. Place the new power of 10 with the decimal in scientific notation form. If your decimal number is greater than 10, count the number of times the decimal moves to the left, and add this number to the exponent.

Division: To divide numbers in scientific notation, first divide the decimal numbers. Then subtract the exponents of your power of 10. Place the new power of 10 with the decimal in scientific notation form. If the resulting decimal number is less than 1, move the decimal point to the right and decrease the exponent by the number of places that the decimal point moved.

 What does it look like?

Express Large Numbers in Standard Form

$$n = 101325 = 1.01325 * 10^5$$ (Move the decimal point 5 places to the right)

Express Small Numbers in Standard Form

$$n = 0.00092 = 9.2 * 10^ {-4}$$ (Move the decimal point 4 places to the left.)

Multiplication:

$$(2.3 * 10^4) * (6.6 * 10^7)$$ First Step - $$2.3 * 6.6 = 16.38$$ Second Step - $$10^4 * 10^7 = 10^{11}$$ Because the new decimal number in step one is greater than 10, count the number of places the decimal moves to put the number between 1 and 10. Add this number to the exponent. In this case, the decimal point moves one place, so add 1 to the exponent. $$1.638 * 10^{12}$$

Division:

$$(1.23 * 10^{10}) ÷ (2.4 * 10^2)$$ First Step - $$1.23 ÷ 2.4 = 0.5125$$ Second Step - (Subtract the exponents of the powers of 10) $$10^{10} ÷ 10^2 = 10^8$$ Because the decimal number is not between 1 and 10, move the decimal point one place to the right and decrease the exponent by 1. $$5.125 * 10^7$$

You'll use it...

It’s used in a lot of places where very large or very small quantities need to be measured.

For example:

The numbers of atoms in a mole (chemistry).

The distances between the planets or stars in the universe, measured in miles.

And in the other extreme, for very tiny numbers, like the size or weight of an atom or molecule.

  Videos

Practice Problems

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Scientific Notation: Addition And Subtraction

The following diagram shows how to add or subtract numbers in scientific notation. Scroll down the page for more examples on adding and subtracting with scientific notation.

To add or subtract two numbers in scientific notation:
Step 1: Adjust the powers of 10 in the 2 numbers so that they have the same index. (Tip: It is easier to adjust the smaller index to equal the larger index).

Step 2: Add or subtract the numbers.

Step 3: Give the answer in scientific notation.

Example:
Evaluate 2 × 103 + 3.6 × 104, giving your answer in scientific notation.

Solution:

Example:
Evaluate 7 × 105 – 5.2 × 104, giving your answer in scientific notation.

Solution:

Add And Subtract Numbers In Scientific Notation With Same Or Different Exponents And Negative Exponents

This video explains how to add and subtract numbers written in scientific notation, whether or not they have the same exponent.

Examples:
(3.769 × 105) + (4.21 × 105)
(8.14 × 10-2) - (2.01 × 10-2)
(7.58 × 105) + (2.871 × 106)
(2.9785 × 10-8) - (5.72 × 10-10)
(4.86 × 103) - (4.72 × 103)

• Show Video Lesson

Why You Have To Make Sure The Exponents Are The Same?

This video explains why when adding and subtracting numbers in scientific notation, you have to make sure the exponents are the same.

Example:
(7.1 × 103) + (5 × 102)

• Show Video Lesson

Rules For Scientific Notation

This video reviews scientific notation and shows how to add and subtract in Scientific Notation.

Rule for scientific notation:

• If the exponent goes down, the decimal point goes right.
• If the decimal point goes right, the exponent goes down.
• If the exponent goes up, the decimal goes left.
• If the decimal point goes left, the exponent goes up.

• Show Video Lesson

Scientific Notation - Addition And Subtraction Rules

This video explains how to do add and subtract in scientific notation without the use of a calculator.

• The powers of 10 must be the same in both terms in order to add/subtract.
• Once the power of 10 matches, just add or subtract.
• Adjust the answer to scientific notation.

• Show Video Lesson

Scientific Notation - Adding And Subtracting

Learn how to add and subtract in scientific notation.
Add and subtract numbers in different bases by rewriting their exponents to be the same.

Example:
(3.5 × 104) + (3.1 × 105)

• Show Video Lesson

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