Powers of 10: multiplying and dividing by 10, 100 and 1,000

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Lesson notes

What is a Power of 10?

A power of 10 is 10 multiplied by itself a certain number of times.
For example, 10 × 10 = 100, so 100 is a power of 10.
The first few powers of 10 are: 1, 10, 100, 1,000, 10,000, 100,000, and so on.
The number 1 is also a power of 10. It is called the zeroth power of 10.
We can write powers of 10 using a small number called an exponent. For example, 10² means 10 × 10, which is 100.
The exponent tells you how many zeros to put after the 1. So 10³ = 1,000 (three zeros).

Multiplying by 10, 100, and 1,000

When you multiply a number by 10, each digit moves one place to the left.
For example, 35 × 10 = 350. The 3 moves from the tens place to the hundreds place, and the 5 moves from the ones place to the tens place.
When you multiply by 100, each digit moves two places to the left.
For example, 35 × 100 = 3,500.
When you multiply by 1,000, each digit moves three places to the left.
For example, 35 × 1,000 = 35,000.
The decimal point does not move. Instead, the digits move around it.

Dividing by 10, 100, and 1,000

When you divide a number by 10, each digit moves one place to the right.
For example, 350 ÷ 10 = 35.
When you divide by 100, each digit moves two places to the right.
For example, 3,500 ÷ 100 = 35.
When you divide by 1,000, each digit moves three places to the right.
For example, 35,000 ÷ 1,000 = 35.
Again, the decimal point stays in the same place; the digits move.

Patterns with Zeros

When you multiply a whole number by 10, you add one zero to the end.
For example, 7 × 10 = 70.
When you multiply by 100, you add two zeros.
For example, 7 × 100 = 700.
When you multiply by 1,000, you add three zeros.
For example, 7 × 1,000 = 7,000.
When you divide a whole number that ends in zeros by 10, you remove one zero.
For example, 7,000 ÷ 10 = 700.

Using Exponent Notation

We can write 10 × 10 as 10². The small 2 is the exponent.
10² = 100, 10³ = 1,000, and 10⁴ = 10,000.
The exponent tells you how many times to multiply 10 by itself.
It also tells you how many zeros are in the answer. For example, 10⁵ has five zeros, so it equals 100,000.
We can also write 10¹ = 10 and 10⁰ = 1.
This notation is very useful for writing very large numbers, like 10⁹ (one billion).

Converting Units of Measure

You can use powers of 10 to convert between units like meters and centimeters.
There are 100 centimeters in 1 meter, so to convert meters to centimeters, multiply by 100.
For example, 3 meters = 3 × 100 = 300 centimeters.
To convert centimeters to meters, divide by 100.
For example, 250 centimeters = 250 ÷ 100 = 2.5 meters.
Similarly, there are 1,000 grams in 1 kilogram, so multiply or divide by 1,000.

Estimating with Powers of 10

You can use powers of 10 to estimate answers to harder calculations.
For example, 29 × 31 is close to 30 × 30, which is 900.
Since 30 = 3 × 10, 30 × 30 = (3 × 10) × (3 × 10) = 9 × 100 = 900.
This trick works because multiplying by 10, 100, or 1,000 is easy.
Estimating helps you check if your exact answer is reasonable.

Negative Powers of 10

Powers of 10 can also be negative, like 10^-1.
10^-1 = 0.1, which is one tenth.
10^-2 = 0.01, which is one hundredth.
10^-3 = 0.001, which is one thousandth.
A negative exponent tells you how many places the decimal point moves to the left.
For example, 10^-4 = 0.0001 (four places).

Scientific Notation

Scientific notation is a way to write very large or very small numbers using powers of 10.
A number in scientific notation looks like this: m × 10ⁿ, where m is a number between 1 and 10.
For example, 5,000 can be written as 5 × 10³.
For very small numbers, like 0.004, we write 4 × 10^-3.
This makes it easier to read and compare huge or tiny numbers.
You will use scientific notation more in later grades, but it is good to know the idea now.

Fun Fact: Googol

A googol is the number 10¹⁰⁰, which is 1 followed by 100 zeros.
The word 'googol' was invented by a 9-year-old boy named Milton Sirotta.
A googol is so big that it is bigger than the number of atoms in the observable universe.
There is an even bigger number called a googolplex, which is 10^(10¹⁰⁰).
That is 10 to the power of a googol, which is unimaginably large.

Slides

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Practice questions

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1.What is 10³ (10 to the power of 3)?
Easy
- A100
- B1000
- C10,000
- D30
2.What is 5.6 × 100?
Easy
- A56
- B560
- C5600
- D0.56
3.Which number is the same as 0.004?
Easy
- A4 × 10^-3
- B4 × 10³
- C4 × 10^-2
- D4 × 10²
4.Which of the following are equal to 1,000? (select all that apply)
Medium
- A10³
- B10 × 100
- C10² × 10
- D1000 × 10
- E10⁴ ÷ 10
5.When you divide a number by 100, the digits move two places to the right.
Easy
True or false?
6.Match each power of 10 to its value.
Medium
- 10²
- 10³
- 10⁴
- 100
- 1,000
- 10,000
7.Put these numbers in order from smallest to largest.
Easy
- 0.05
- 5 × 10^-2
- 5 ÷ 100
- 0.5
8.What is 0.07 ÷ 100?
Medium
- A0.0007
- B0.007
- C0.7
- D7

Revision notes

## Powers of 10: Multiplying and Dividing by 10, 100, and 1,000 When you multiply or divide a number by 10, 100, or 1,000, the digits move places, but the decimal point stays in the same spot. This is a quick way to make numbers bigger or smaller. ### Multiplying by 10, 100, 1,000 - **Multiplying by 10** moves each digit one place to the left. The number gets 10 times bigger. - Example: 3.45 × 10 = 34.5 (the digits move left, so 3 becomes tens, 4 becomes ones, 5 becomes tenths) - **Multiplying by 100** moves each digit two places to the left. The number gets 100 times bigger. - Example: 3.45 × 100 = 345 - **Multiplying by 1,000** moves each digit three places to the left. The number gets 1,000 times bigger. - Example: 3.45 × 1,000 = 3,450 (you may need to add zeros to hold the places) **Tip:** When multiplying by 10, 100, or 1,000, you can just move the decimal point to the right by the same number of zeros: ×10 → move 1 place, ×100 → move 2 places, ×1,000 → move 3 places. ### Dividing by 10, 100, 1,000 - **Dividing by 10** moves each digit one place to the right. The number gets 10 times smaller. - Example: 345 ÷ 10 = 34.5 - **Dividing by 100** moves each digit two places to the right. The number gets 100 times smaller. - Example: 345 ÷ 100 = 3.45 - **Dividing by 1,000** moves each digit three places to the right. The number gets 1,000 times smaller. - Example: 345 ÷ 1,000 = 0.345 **Tip:** When dividing, move the decimal point to the left by the same number of zeros: ÷10 → move 1 place, ÷100 → move 2 places, ÷1,000 → move 3 places. ### Powers of 10 and Exponents A **power of 10** is 10 multiplied by itself a certain number of times. We write it using an **exponent** (a small number that tells how many times to multiply 10 by itself). - 10^1 = 10 (one zero) - 10^2 = 100 (two zeros) - 10^3 = 1,000 (three zeros) - 10^4 = 10,000 (four zeros) So, 10^n is a 1 followed by n zeros. **Negative exponents** show fractions or decimals: - 10^{-1} = 0.1 (one decimal place) - 10^{-2} = 0.01 (two decimal places) - 10^{-3} = 0.001 (three decimal places) ### Using Powers of 10 to Convert Units You can use powers of 10 to change between units like meters and kilometers, or grams and kilograms. - 1 kilometer = 1,000 meters (so multiply by 10^3 to convert km to m) - 1 meter = 0.001 kilometers (so divide by 10^3 to convert m to km) ### Estimating with Powers of 10 Powers of 10 help you estimate answers. For example, 4,893 × 10 is about 5,000 × 10 = 50,000. This gives a quick, rough answer. ### Key Points to Remember - The decimal point never moves; the digits move. - Multiplying by 10, 100, 1,000 makes numbers bigger (move decimal right). - Dividing by 10, 100, 1,000 makes numbers smaller (move decimal left). - 10^n = 1 followed by n zeros. - 10^{-n} = 0.000...1 with n decimal places. Practice with these examples: - 6.7 × 100 = 670 - 0.05 × 1,000 = 50 - 4,500 ÷ 100 = 45 - 0.8 ÷ 10 = 0.08

Power of 10: A number that can be written as 10 multiplied by itself a certain number of times. For example, 10^2 = 100, 10^3 = 1,000.
Exponent: A small number written above and to the right of a base number that tells how many times to multiply the base by itself. In 10^3, the exponent is 3.
Decimal point: The dot in a number that separates the whole number part from the fractional part. For example, in 3.45, the decimal point is between 3 and 4.
Place value: The value of a digit based on its position in a number. For example, in 345, the 3 is in the hundreds place, so it is worth 300.
Multiplying by 10, 100, 1,000: A shortcut to make a number 10, 100, or 1,000 times bigger by moving the digits to the left (or moving the decimal point to the right) by 1, 2, or 3 places.
Dividing by 10, 100, 1,000: A shortcut to make a number 10, 100, or 1,000 times smaller by moving the digits to the right (or moving the decimal point to the left) by 1, 2, or 3 places.
Negative exponent: An exponent that is a negative number, which means the power of 10 is a fraction or a decimal. For example, 10^{-2} = 0.01.
Scientific notation: A way to write very large or very small numbers using a number between 1 and 10 multiplied by a power of 10. For example, 3,000,000 can be written as 3 × 10^6.

Flashcards

What does 'power of 10' mean?

A power of 10 is 10 multiplied by itself a certain number of times. For example, 10² = 10 × 10 = 100.

What is 10⁰?

10⁰ = 1. By definition, any number to the power of 0 is 1.

How do you write 10³ as a number?

10³ = 1,000. It is 1 followed by 3 zeros.

What is 10⁵?

10⁵ = 100,000. It is 1 followed by 5 zeros.

What is 10^-1?

10^-1 = 0.1. It is 1 divided by 10.

What is 10^-3?

10^-3 = 0.001. It is 1 divided by 1,000.

When you multiply a number by 10, what happens to the digits?

The digits move one place to the left. For example, 35 × 10 = 350.

When you multiply a number by 100, what happens to the digits?

The digits move two places to the left. For example, 35 × 100 = 3,500.

When you multiply a number by 1,000, what happens to the digits?

The digits move three places to the left. For example, 35 × 1,000 = 35,000.

When you divide a number by 10, what happens to the digits?

The digits move one place to the right. For example, 350 ÷ 10 = 35.

When you divide a number by 100, what happens to the digits?

The digits move two places to the right. For example, 3,500 ÷ 100 = 35.

When you divide a number by 1,000, what happens to the digits?

The digits move three places to the right. For example, 35,000 ÷ 1,000 = 35.

What is 2.5 × 10?

2.5 × 10 = 25. The digits move one place left.

What is 2.5 × 100?

2.5 × 100 = 250. The digits move two places left.

What is 2.5 × 1,000?

2.5 × 1,000 = 2,500. The digits move three places left.

What is 250 ÷ 10?

250 ÷ 10 = 25. The digits move one place right.

What is 250 ÷ 100?

250 ÷ 100 = 2.5. The digits move two places right.

What is 250 ÷ 1,000?

250 ÷ 1,000 = 0.25. The digits move three places right.

How many zeros are in the product of 10 × 10?

10 × 10 = 100, which has 2 zeros. The number of zeros is the sum of the zeros in the factors (1+1=2).

How many zeros are in 10⁴?

10⁴ = 10,000, which has 4 zeros. The exponent tells you the number of zeros.

What is 10² × 10³?

10² × 10³ = 10²⁺³ = 10⁵ = 100,000. When multiplying powers of 10, add the exponents.

What is 10⁵ ÷ 10²?

10⁵ ÷ 10² = 10^5-2 = 10³ = 1,000. When dividing powers of 10, subtract the exponents.

How do you write 0.001 as a power of 10?

0.001 = 10^-3. The negative exponent shows how many places the decimal point moves.

What is the number called that is 10¹⁰⁰?

It is called a googol. It is 1 followed by 100 zeros.

What is scientific notation?

Scientific notation writes a number as a number between 1 and 10 multiplied by a power of 10. For example, 5,000 = 5 × 10³.

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