Express Numbers Using Powers of 10

Students write very large and very small numbers in the form a × 10ⁿ where 1 ≤ a < 10, and convert between standard form and this single-digit-times-power-of-10 form. They understand how the exponent reflects the magnitude of the number and use this to compare sizes.

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

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20 rungs. A star at the top.

1000 = 1 × 10

How it is taught

0:00 of 4:14

A power of 10 counts how many places the point moves.

A number can be written as a digit times a power of 10.

Read it: Powers of ten as places

0.032 = 3.2 × 10^−2: from 3.2 to 0.032 the digits slide 2 places right, so the power is −2. Each place is one power of ten.

Write both numbers in the columns of places.
Count the places the digits slide, one hop a place.
Left makes a positive power; right makes a negative one.

Each place left is 10 times bigger, and each place right is 10 times smaller.

So 2 places right is 10 × 10 times smaller: 10^−2.

Count places, not zeros: 3600 has 2 zeros, but the digits of 3.6 slide 3 places to make it.

Each place the digits slide is one power of ten: left is positive, right is negative.

Counting the zeros: 3600 has 2 zeros, so writing 3.6 × 10^2.

Count the places the digits slide: from 3.6 to 3600 is 3 places left, so 3.6 × 10^3.

8000000 = 8 × 10^___ — 6. The digits slide 6 places left from 8 to 8000000: 10^6.

4,500,000 = 4.5 × 10⁶

Imagine a city with four million, five hundred thousand people. Writing all those zeros gets messy fast!

Here's our number: 4,500,000 — seven whole digits long.

Let's rewrite it as a single digit times a power of ten. That's called scientific notation.

First, slide the decimal point left until only one digit sits in front of it.

Count how many places we moved it. One, two, three, four, five, six places.

That six becomes the exponent. We write four point five times ten to the sixth.

Ten to the sixth means ten multiplied by itself six times — that equals one million.

And four point five times one million is 4,500,000. It checks out!

The exponent tells us the size. A 6 means millions. A 9 would mean billions.

Grade 8 · Writing big numbers with powers of 10

So 4,500,000 written with a power of ten is 4.5 × 10⁶. Shorter — and the exponent shouts the size!

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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