Represent very large and small metric units using powers of ten

Ontario Grade 8 Mathematics

Spatial Sense

Representing very large and very small metric units

A kilometre and a millimetre measure length, but they describe very different sizes. Metric prefixes help us name those sizes. Powers of ten give us a compact way to show how each prefixed unit compares with its base unit. In this lesson, the metre is the base unit for length. The same prefix relationships work with other metric base units, such as the gram and the litre.

What you will learn

Recognize how selected metric prefixes relate to the base unit.
Write large and small metric units using powers of ten.
Convert between a prefixed metric unit and its base unit using powers of ten.

1. Start with metric units

A metre is a metric unit of length. A prefix is a word part placed before a unit name. The prefix tells how the unit compares with the base unit. For example, kilo- means one thousand times the base unit. So one kilometre is one thousand metres.

A power of ten is a way to write repeated multiplication or division by ten. The small raised number is called an exponent. For example,

10^3

means

10 \times 10 \times 10

, which equals

1000

. A negative exponent represents a small fraction:

10^{-3}

means

1 \div 10 \div 10 \div 10

, which equals

0.001

.

You can picture the prefix before writing symbols. One kilometre is a group of one thousand metres. One millimetre is one thousandth of a metre. The powers of ten make those relationships shorter to write.

10^3=1000,\quad 10^{-3}=0.001

A prefix changes the size of a metric unit.
A positive exponent such as $3$ can represent a large number of base units.
A negative exponent such as $-3$ can represent a small part of a base unit.

2. Connect common prefixes to powers of ten

The base unit has no prefix in its name. For length, the base unit is the metre. A kilometre is larger than a metre, while a centimetre and a millimetre are smaller.

Use the prefix relationships below as a guide. The symbol m can stand for metre, but it can also appear in a prefix symbol. In km, the k is the prefix symbol and m is the unit symbol. Read km as kilometre.

The expression

1\text{ km}=10^3\text{ m}

says that one kilometre is the same length as one thousand metres. The expression

1\text{ mm}=10^{-3}\text{ m}

says that one millimetre is one thousandth of a metre.

1\text{ km}=10^3\text{ m},\quad 1\text{ cm}=10^{-2}\text{ m},\quad 1\text{ mm}=10^{-3}\text{ m},\quad 1\text{ µm}=10^{-6}\text{ m}

kilo- (k) means $10^3$ times the base unit.
centi- (c) means $10^{-2}$ times the base unit.
milli- (m) means $10^{-3}$ times the base unit.
micro- (µ) means $10^{-6}$ times the base unit.

3. Read and write the representation

A measurement can be written in a prefixed unit or in a base unit. To express a prefixed unit using powers of ten, replace the prefix with its power-of-ten relationship. Keep the measured amount and the unit clear.

For example,

7\text{ km}

means seven groups of one thousand metres. Using a power of ten, it is

7\times10^3\text{ m}

. The number 7 is the amount, and

10^3

represents the size of each kilometre in metres.

A small amount works in the same way. Since one millimetre is

10^{-3}

of a metre,

5\text{ mm}

is

5\times10^{-3}\text{ m}

. The negative exponent tells us that each millimetre is only a small part of a metre.

a\text{ km}=a\times10^3\text{ m},\quad a\text{ mm}=a\times10^{-3}\text{ m}

Keep the measured amount in front of the power of ten.
Use the base unit after the power when expressing a prefixed unit in base units.
A negative exponent does not mean the measurement is negative.

4. Use the relationship to convert

To convert from a prefixed unit to the base unit, use the prefix relationship. For kilometres to metres, multiply the number of kilometres by

10^3

. For millimetres to metres, multiply by

10^{-3}

, which is the same as taking one thousandth of the number.

A quick size check helps. A number of kilometres should become a larger number of metres. A number of millimetres should become a smaller number of metres. This check helps catch a reversed power or an incorrect sign on the exponent.

You can also write a base-unit measurement with a prefix when the amount fits. For example,

0.004\text{ m}

is four thousandths of a metre, so it can be represented as

4\text{ mm}

. The power-of-ten form shows why the two notations match.

1\text{ mm}=10^{-3}\text{ m}=0.001\text{ m}

Kilometres to metres uses the factor $10^3$ .
Millimetres to metres uses the factor $10^{-3}$ .
Check that the result has the expected size and the correct unit.

Selected metric prefixes for length

Unit name	Symbol	Relationship to the metre
kilometre	km	$1\text{ km}=10^3\text{ m}$
centimetre	cm	$1\text{ cm}=10^{-2}\text{ m}$
millimetre	mm	$1\text{ mm}=10^{-3}\text{ m}$
micrometre	µm	$1\text{ µm}=10^{-6}\text{ m}$

Worked example

Write a large metric unit using a power of ten

Represent

6.4\text{ km}

in metres using a power of ten.

Use the prefix relationship
The prefix kilo- means one thousand times the base unit. Replace each kilometre with $10^3$ metres.
$1\text{ km}=10^3\text{ m}$
Keep the measured amount
There are $6.4$ kilometres, so multiply $6.4$ by the power of ten. The result is still expressed in metres.
$6.4\text{ km}=6.4\times10^3\text{ m}$
Check the size
The metre amount is much larger than $6.4$ because each kilometre contains one thousand metres. The representation has the expected size.
$6.4\times10^3=6400$

Answer:

6.4\text{ km}=6.4\times10^3\text{ m}

Check: Since

10^3=1000

, this is

6400\text{ m}

, which is larger than

6.4\text{ km}

as a number of metres should be.

Worked example

Write a small metric unit using a power of ten

Represent

8\text{ mm}

in metres using a power of ten.

Use the prefix relationship
The prefix milli- means one thousandth of the base unit. So each millimetre is $10^{-3}$ of a metre.
$1\text{ mm}=10^{-3}\text{ m}$
Represent eight millimetres
There are eight millimetres. Use eight groups of one thousandth of a metre.
$8\text{ mm}=8\times10^{-3}\text{ m}$
Check the size
One thousandth is a small part of a metre, so eight millimetres must be less than one metre. The decimal form confirms this.
$8\times10^{-3}=0.008$

Answer:

8\text{ mm}=8\times10^{-3}\text{ m}

Check: The decimal form is

0.008\text{ m}

, which is less than one metre.

Worked example

Connect a decimal measurement to a prefix

Express

0.025\text{ m}

in centimetres, then represent the same length using a power of ten in metres.

Recall the centimetre relationship
One centimetre is one hundredth of a metre. In the other direction, one metre contains one hundred centimetres.
$1\text{ cm}=10^{-2}\text{ m}$
Find the centimetre amount
Since each centimetre is $0.01$ metre, $0.025$ metre is two and a half centimetres. This follows because $2.5$ groups of $0.01$ metre make $0.025$ metre.
$0.025\text{ m}=2.5\text{ cm}$
Write the power-of-ten form
To show the original length in metres using a power of ten, write the decimal amount as a number of thousandths of a metre.
$0.025\text{ m}=25\times10^{-3}\text{ m}$

Answer:

2.5\text{ cm}

and

25\times10^{-3}\text{ m}

Check: Since

25\times0.001=0.025

, the power-of-ten form matches the original metre measurement.

Common mistakes and how to avoid them

Writing a small metric unit with a positive exponent, such as

1\text{ mm}=10^3\text{ m}

.

Correction: A millimetre is one thousandth of a metre, so use the negative exponent:

1\text{ mm}=10^{-3}\text{ m}

.

Thinking the negative exponent makes the measured amount negative.

Correction: The exponent shows a small fraction of the base unit. A measurement such as

8\times10^{-3}\text{ m}

is positive.

Changing the unit name but leaving out the prefix relationship.

Correction: Include the power of ten when expressing a prefixed unit in base units, such as

3\text{ km}=3\times10^3\text{ m}

.

Lesson summary

Metric prefixes show how a unit compares with its base unit.
Kilo- represents $10^3$ times the base unit; centi-, milli-, and micro- represent $10^{-2}$ , $10^{-3}$ , and $10^{-6}$ of the base unit.
A negative exponent represents a fraction of the base unit, not a negative measurement.
Keep the amount, power of ten, and unit together when writing a representation.

Check your understanding

Question 1

Which expression represents

4\text{ km}

in metres?

$4\times10^{-3}\text{ m}$
$4\times10^3\text{ m}$
$4\times10^2\text{ m}$
$4\times10^{-2}\text{ m}$

Show answer and explanation

4\times10^3\text{ m}

Kilo- means one thousand, and

1000=10^3

. Therefore

4\text{ km}=4\times10^3\text{ m}

.

Question 2

Which expression represents

3\text{ mm}

in metres?

$3\times10^3\text{ m}$
$3\times10^{-2}\text{ m}$
$3\times10^{-3}\text{ m}$
$3\times10^{-6}\text{ m}$

Show answer and explanation

3\times10^{-3}\text{ m}

Milli- means one thousandth, which is

10^{-3}

. So

3\text{ mm}=3\times10^{-3}\text{ m}

.

Question 3

A measurement is

0.06\text{ m}

. Which is an equivalent centimetre measurement?

$0.6\text{ cm}$
$6\text{ cm}$
$60\text{ cm}$
$600\text{ cm}$

Show answer and explanation

6\text{ cm}

One centimetre is

0.01

metre. Six groups of

0.01

metre make

0.06

metre, so the equivalent measurement is

6\text{ cm}

.

Key terms

Base unit: The main unit used to compare metric units of the same kind. The metre is the base unit used for length here.
Prefix: A word part before a unit name that tells how the unit compares with the base unit.
Power of ten: A compact way to write a number made from repeated factors of ten, or a fraction made by dividing by ten repeatedly.
Exponent: The small raised number that tells how many factors of ten are used. A negative exponent represents repeated division by ten.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation E2.1. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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E2.1 · Represent very large and small metric units using powers of ten

Ontario Grade 8 Mathematics

Representing very large and very small metric units

What you will learn

1. Start with metric units

2. Connect common prefixes to powers of ten

3. Read and write the representation

4. Use the relationship to convert

Selected metric prefixes for length

Write a large metric unit using a power of ten

Write a small metric unit using a power of ten

Connect a decimal measurement to a prefix

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 8 Math

About this lesson and its review