Multiply and Divide Numbers in Scientific Notation

Students multiply and divide numbers written in scientific notation by separately operating on the coefficients and the powers of 10, then adjusting the result back to proper scientific notation form. They interpret computed products and quotients in context (e.g., comparing planetary distances).

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Grade 8 · Expressions and Equations: Exponents and Scientific Notation · question 1 of 3

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6 × 10 = (1.5 × 10^2) × (4 × 10^1)

How it is taught

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Multiply the front numbers, and add the exponents on 10.

Then slide the point so the front number is at least 1 and less than 10.

Read it: Scientific notation: × and ÷

(2 × 10^3) × (4 × 10^5) multiplies in two parts. The numbers: 2 × 4 = 8. The powers: 10^3 × 10^5 = 10^8, because the exponents add. So the product is 8 × 10^8.

Multiply or divide the numbers in front.
Add the exponents to multiply the powers, or take them away to divide.
Write the answer as the number in front times the power of ten.

Multiplication can be done in any order, so the numbers and the powers of ten can be worked apart.

10^3 × 10^5 is 3 tens multiplied by 5 more tens: 8 tens in all, 10^8.

Dividing undoes multiplying, so 10^8 ÷ 10^5 takes the 5 tens away again: 10^3.

Multiply: multiply the numbers and add the exponents. Divide: divide the numbers and take the exponents away.

Multiplying the exponents: 10^3 × 10^5 = 10^15.

The exponents count tens, so they add: 3 + 5 = 8, 10^8.

(4 × 10^7) × (2 × 10^4) = 8 × 10^___ — 11. Multiply the numbers, 4 × 2 = 8, and the powers: the exponents add, 7 + 4 = 11.

(3 × 10⁸) × (2 × 10⁵)

Let's multiply two numbers in scientific notation: three times ten to the eighth, times two times ten to the fifth.

Scientific notation has two parts: a coefficient and a power of ten. We'll handle them separately.

First, group the coefficients together: three times two.

Then group the powers of ten: ten to the eighth times ten to the fifth.

Multiply the coefficients. Three times two is six.

When we multiply powers with the same base, we add the exponents.

Grade 8 · Multiply coefficients, add exponents

Eight plus five is thirteen. So ten to the eighth times ten to the fifth is ten to the thirteenth.

Put it back together: six times ten to the thirteenth.

Check the form. The coefficient must be at least one and less than ten. Six fits — we're good!

So three times ten to the eighth, times two times ten to the fifth, equals six times ten to the thirteenth. Nice work!

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics

Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal: 1 star — one skill mastered, every rung of its ladder climbed
Climbing now: Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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