B. Number · Number Sense: Big Numbers, Square Roots and Rational Numbers

Expanded Form with Powers of Ten

B1.1About 40 minutes

Learning goals

I can write a power of ten, like 10⁶, in standard form.
I can explain what the base and the exponent mean in a power of ten.
I can write a big number in expanded form using powers of ten.
I can change expanded form with powers of ten back to standard form.

Words to know

power of ten
A number made by multiplying 10 by itself, like 10 × 10 × 10 = 1000. It can be written as 10³.

base
In 10³, the 10. It is the number being multiplied.

exponent
In 10³, the small raised 3. It tells how many 10s are multiplied together.

expanded form
A number written as the sum of the values of its digits, like 4 × 10⁶ + 6 × 10⁵.

Remember

In the last lesson you saw that each place is worth 10 times the place to its right. Writing all those zeros gets tiring! Mathematicians use a short way to write numbers like 1000, 1 000 000 and 1 000 000 000: powers of ten.

Counting tens

100 = 10 × 10. Two tens are multiplied, so we write 10² ("ten to the power of two" or "ten squared").
1000 = 10 × 10 × 10 = 10³ ("ten to the power of three" or "ten cubed").
1 000 000 = six 10s multiplied = 10⁶.

In 10³, the 10 is the base and the small raised 3 is the exponent. The exponent tells how many 10s are multiplied together.

Here is a handy pattern: for powers of ten, the exponent equals the number of zeros. 10⁶ has 6 zeros: 1 000 000.

(In the Operations unit you will use exponents with other bases too, like 2⁵.)

In 10³, the base is 10 and the exponent 3 says how many 10s are multiplied. For powers of ten, the exponent also matches the number of zeros.

Power of ten	Repeated multiplication	Standard form	Name
10⁰	(no 10s)	1	one
10¹	10	10	ten
10²	10 × 10	100	hundred
10³	10 × 10 × 10	1000	thousand
10⁴	10 × 10 × 10 × 10	10 000	ten thousand
10⁵	five 10s	100 000	hundred thousand
10⁶	six 10s	1 000 000	million
10⁷	seven 10s	10 000 000	ten million
10⁸	eight 10s	100 000 000	hundred million
10⁹	nine 10s	1 000 000 000	billion

Each power of ten is 10 times the one above it in the table.

Why is 10⁰ = 1?

Go up the table: each time the exponent goes down by 1, you divide by 10. 10² = 100, 10¹ = 10, so 10⁰ = 10 ÷ 10 = 1. That is why the ones place can be written as 10⁰. You can also just write the ones digit by itself.

Expanded form with powers of ten

Each place in our number system is a power of ten. So a big number can be written as digit × power of ten, added together. Leave out any place that has a 0.

3 405 000 = 3 000 000 + 400 000 + 5000 = 3 × 10⁶ + 4 × 10⁵ + 5 × 10³

Each non-zero digit of 3 405 000 becomes digit × power of ten. The places with 0 are skipped.

Worked example

Example 1: Power to standard form

Write 7 × 10⁵ in standard form.

10⁵ = 100 000 (5 zeros).
7 × 100 000 = 700 000.

7 × 10⁵ = 700 000

Worked example

Example 2: Standard form to expanded form

Write 52 030 000 in expanded form using powers of ten.

Find the non-zero digits and their places: 5 is in the ten millions place (10⁷), 2 is in the millions place (10⁶), 3 is in the ten thousands place (10⁴).
Write each as digit × power of ten and add them.

52 030 000 = 5 × 10⁷ + 2 × 10⁶ + 3 × 10⁴

Worked example

Example 3: Expanded form to standard form

Write 6 × 10⁸ + 9 × 10⁶ + 4 × 10² + 8 in standard form.

6 × 10⁸ = 600 000 000
9 × 10⁶ = 9 000 000
4 × 10² = 400, and the last part is 8 ones.
Add: 600 000 000 + 9 000 000 + 400 + 8. Places with no term get a 0.

609 000 408

Worked example

Example 4: A headline number

A news story says a project will cost $2.3 million. Write the amount in standard form and in expanded form.

2.3 million = 2 million + 0.3 million. 0.3 million = 3 tenths of a million = 300 000.
So $2.3 million = $2 300 000.
Expanded: 2 × 10⁶ + 3 × 10⁵.

$2 300 000 = 2 × 10⁶ + 3 × 10⁵. News reports often use decimals of a million because they are short and easy to read.

Hands-on activity

Powers of ten around the house

You need: paper, a pencil, and things with metric labels (a ruler or measuring tape, a food package that shows grams, a drink container that shows mL or L).

Fill in these with powers of ten. Use the labels and your ruler to help.

1 m = ___ mm (answer: 1000 = 10³)
1 km = ___ m (answer: 1000 = 10³)
1 km = ___ mm (answer: 1 000 000 = 10⁶)
1 kg = ___ g (answer: 1000 = 10³)
1 L = ___ mL (answer: 1000 = 10³)

Write the digits 0 to 9 on sticky notes, plus eight extra 0s (big numbers need lots of zeros). Ask your grown-up for a big number, like "forty-two million eight hundred thousand". Build it, then write it in expanded form with powers of ten (4 × 10⁷ + 2 × 10⁶ + 8 × 10⁵).
Switch: you write an expanded form and your grown-up writes the standard form. Check each other.

Math talk: "How did you know which power of ten to use?"

Just for fun

A googol is 10¹⁰⁰: a 1 followed by one hundred zeros. The company name Google comes from a playful misspelling of that word. Imagine writing that cheque!

Your turn

Write 10⁴ in standard form.

Write 1 000 000 as a power of ten.

Write 8 × 10⁷ in standard form.

Write 4 600 000 in expanded form using powers of ten.

Write 2 × 10⁸ + 5 × 10⁵ + 1 × 10³ in standard form.

What does the exponent tell you in 10⁵?

Write 30 070 900 in expanded form using powers of ten.

More practice: warm-up

Write 10³ in standard form.

Write 10 × 10 × 10 × 10 × 10 × 10 as a power of ten.

What is 10⁹ in standard form? What is its name?

Write 6 × 10² in standard form.

Write 100 as a power of ten.

More practice: on your own

Write 905 000 060 in expanded form using powers of ten.

Write 4 × 10⁸ + 3 × 10⁷ + 2 × 10⁰ in standard form.

Write 3.6 million in standard form and in expanded form.

Which is greater, 5 × 10⁶ or 9 × 10⁵? Explain.

Ella says 10⁴ = 40. What mistake did she make?

Challenge

How many times greater is 10⁹ than 10⁶?

Mo wrote 12 × 10⁵. Write it in standard form, then fix the expanded form so each digit is from 1 to 9.

Write 250 million in standard form, in expanded form with powers of ten, and as a decimal number of billions.

Check your understanding

Quiz

What is 10⁵ in standard form?

Which power of ten equals one million?

What is 4 × 10⁷?

Which is the expanded form of 2 030 000?

In 10³, what is the 3 called?

What is 6 × 10⁸ + 5 × 10³ in standard form?

For parents and teachers

Big idea: Ontario's Grade 7 curriculum (B1.1) asks students to represent whole numbers up to one billion, including in expanded form using powers of ten. Exponents in general (like 2⁵ or 3⁴) are taught in the Operations unit (B2.7). Here we use only powers of ten.

Common difficulties:

Multiplying the base by the exponent (10⁴ = 40). Have your child say "four tens multiplied together" aloud.
Using the wrong power for a place. Counting the places to the right of the digit gives the exponent: in 52 030 000, the 5 has 7 digits after it, so it is 5 × 10⁷.
Forgetting placeholder zeros when going back to standard form.
Reading 3.6 million as 3 600. Remind your child that 0.1 million is 100 000.

How to help: Point out metric facts (1 km = 10³ m, 1 kg = 10³ g). When you see headlines like "$4.5 million", ask your child to write the full number.