Year 7NumberNumber structures and operationsYear 7 Maths

Powers of 10 and expanded form

What this lesson covers

  • In our number system each place value is a power of 10, and this continues infinitely.
  • Repeated multiplication can be expressed using exponent notation with positive exponents.
  • An exponent means 'raising to the power of' (e.g. 5² is 5 raised to the power of 2 or 5 to the second power).
  • Expanded form uses powers of 10 to indicate place value.
Key idea

Ten to the power of four is not forty

  • It is a very easy mistake to make.
  • The little 4 is not something to multiply by.
  • It is a count.
WrittenMeans
10⁴10 × 10 × 10 × 10
not10 × 4
Key idea

What the exponent counts

  • The big number is the base: the one being multiplied.
  • The small raised number is the exponent.
  • It says how many of the base to multiply together.
ShortLongValue
2³2 × 2 × 28
Key idea

How to say it out loud

  • 2³ is "two to the power of three", or "two cubed".
  • 5² is "five squared": the power of two.
  • Those words come from area and volume.
ShortLongValue
2³2 × 2 × 28
5²5 × 525
Key idea

Any base at all

  • The base does not have to be small.
  • 3⁴ is four threes multiplied: 81.
  • Not 3 × 4 = 12. Nowhere near it.
ShortLongValue
2³2 × 2 × 28
5²5 × 525
3⁴3 × 3 × 3 × 381
10³10 × 10 × 101000
Key idea

Ten is the useful one

  • Multiply tens together and something tidy happens.
  • 10 × 10 × 10 is 1000.
  • Three tens multiplied. Three zeros. Count them.
ShortLongValue
10³10 × 10 × 101000
Key idea

The powers of ten

  • The exponent and the number of zeros are the same.
  • That is only true for base ten.
  • 3⁴ is 81; no zeros anywhere.
PowerValueColumn
10¹10tens
10²100hundreds
10³1000thousands
10⁴10,000ten thousands
10⁵100,000hundred thousands
10⁶1,000,000millions
Key idea

Every column is a power of ten

  • Look at the third column of that table.
  • Ones, tens, hundreds, thousands: the place value chart.
  • It was powers of ten the whole time.
PowerValueColumn
10¹10tens
10²100hundreds
10³1000thousands
10⁴10,000ten thousands
10⁵100,000hundred thousands
10⁶1,000,000millions
Key idea

So what is the ones column?

  • Come down the table: 1000, then 100, then 10.
  • Each step divides by ten, and the exponent drops by one.
  • One more step: 10⁰ is 1.
PowerValueColumn
10⁰1ones
10¹10tens
10²100hundreds
10³1000thousands
10⁴10,000ten thousands
10⁵100,000hundred thousands
10⁶1,000,000millions
Key idea

And it never stops

  • Millions is 10⁶. Billions is 10⁹.
  • There is no biggest column.
  • Every one of them is ten times the one before.
PowerValueColumn
10⁴10,000ten thousands
10⁵100,000hundred thousands
10⁶1,000,000millions
Key idea

A number in its columns

  • Thirty-four thousand, five hundred and six.
  • Each digit sits in a column.
  • Nothing changed except what we call the columns.
A number in its columns
Key idea

You already know expanded form

  • Saying the number aloud names the columns.
  • "Thirty-four thousand" is 3 ten-thousands and 4 thousands.
  • Expanded form writes exactly that down.
Way of writing it34,506
In wordsthirty-four thousand, five hundred and six
Plain sum30,000 + 4,000 + 500 + 0 + 6
With powers3 × 10⁴ + 4 × 10³ + 5 × 10² + 0 × 10¹ + 6 × 10⁰
Key idea

The column with nothing in it

  • 34,506 has no tens.
  • It still has a tens column.
  • Write 0 × 10¹, or the columns after it shift.
The column with nothing in it
Worked example

Write 60,208 in expanded form

  • Sixty thousand, two hundred and eight.
  • Ten-thousands 6, thousands 0, hundreds 2, tens 0, ones 8.
  • 6 × 10⁴ + 0 × 10³ + 2 × 10² + 0 × 10¹ + 8 × 10⁰.
Write 60,208 in expanded form
Worked example

Check it by adding up

  • 6 × 10,000 is 60,000.
  • 2 × 100 is 200. 8 × 1 is 8.
  • 60,000 + 200 + 8 = 60,208. It matches.
TermValue
6 × 10⁴60,000
0 × 10³0
2 × 10²200
0 × 10¹0
8 × 10⁰8
Total60,208
Key idea

Comparing with powers

  • 9,999 has four digits. 10⁴ has five.
  • More columns means bigger, always.
  • So 9,999 is less than 10⁴, by one.
NumberDigitsWhich
9,999four digitssmaller
10⁴ = 10,000five digitsbigger
Worked example

Put these in order

  • 10³, 998, 10⁴, 9,999.
  • Turn the powers into numbers first: 1000 and 10,000.
  • 998, then 10³, then 9,999, then 10⁴.
In orderValue
998998
10³1,000
9,9999,999
10⁴10,000
Key idea

The mistake to watch for

  • 5² is not 10. It is 5 × 5, which is 25.
  • 10⁴ is not 40. It is 10,000.
  • The exponent counts. It never multiplies.
Looks likeActually
5² = 105² = 25
10⁴ = 4010⁴ = 10,000
Your turn

Your turn: four questions

  • Pause here and try all four.
  • Question 4 runs the process backwards.
  • Watch the columns on that one.
Question
1Write 2⁵ as a multiplication, then work it out.
2Which is bigger: 9,999 or 10⁴?
3Write 40,703 in expanded form using powers of 10.
4What number is 5 × 10³ + 2 × 10¹ + 9 × 10⁰?
Show answer

1. 2 × 2 × 2 × 2 × 2 = 32. 2. 10⁴ = 10,000, which is bigger than 9,999 by one. 3. 4 × 10⁴ + 0 × 10³ + 7 × 10² + 0 × 10¹ + 3 × 10⁰. 4. 5000 + 20 + 9 = 5029. Note the empty hundreds column.

Worked example

The answers

  • 2⁵ = 32. And 10⁴ is bigger than 9,999.
  • 40,703 is 4 × 10⁴ + 0 × 10³ + 7 × 10² + 0 × 10¹ + 3 × 10⁰.
  • The last one is 5029: no hundreds column.
Answer
12 × 2 × 2 × 2 × 2 = 32
210⁴ = 10,000, bigger by one
34 × 10⁴ + 0 × 10³ + 7 × 10² + 0 × 10¹ + 3 × 10⁰
45000 + 20 + 9 = 5029
Summary

What to take away

  • An exponent counts factors; it does not multiply.
  • Every place value column is a power of ten.
  • Expanded form is the number said aloud, written in symbols.
Remember
An exponentcounts how many are multiplied
10⁴10 × 10 × 10 × 10 = 10,000
Every columnis a power of 10
10⁰is 1: the ones column
Expanded formis the number said aloud, in symbols

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