Powers of Ten and Exponents | Grade 5 Math | LevelKite

You already know that each place in a number is worth 10 times the place to its right. That fact lets you build large numbers out of tens.

10 tens make 1 hundred, so $10 \times 10 = 100$ .
10 hundreds make 1 thousand, so $10 \times 100 = 1{,}000$ . Since 100 is $10 \times 10$ , you can write this with tens only: $10 \times 10 \times 10 = 1{,}000$ .

A factor is one of the numbers you multiply together. The answer is the product. In $10 \times 10 \times 10$ , every factor is 10. When every factor in a product is 10, the result is a power of ten.

So 100 is a power of ten with 10 as a factor 2 times. And 1,000 is a power of ten with 10 as a factor 3 times. The number 10 is a power of ten too. It has just one factor of 10.

Products with many tens take a while to write, and it is easy to lose count. There is a shorter way.

Write 10 once. Then write a small raised number just after it. That number tells how many times 10 is a factor. The small raised number is the exponent. Because it counts factors, the exponent here is a whole number such as 1, 2 or 3.

For example, $10 \times 10 \times 10$ has 3 factors of 10, so you write it as $10^3$ . Read $10^3$ aloud as “ten to the third power.” Some people say “ten to the power of three.” Both mean the same thing.

\underbrace{10 \times 10 \times 10}_{3 \text{ factors of } 10} = 10^{3}

Four powers of ten: the factors, the exponent, and how to say it

Factors written out	How many factors of 10	With an exponent	Say it aloud
$10$	1	$10^1$	ten to the first power
$10 \times 10$	2	$10^2$	ten to the second power
$10 \times 10 \times 10$	3	$10^3$	ten to the third power
$10 \times 10 \times 10 \times 10$	4	$10^4$	ten to the fourth power

Worked example

Write a product of tens with an exponent

Write $10 \times 10 \times 10 \times 10 \times 10$ with an exponent. Then tell how to read it aloud.

Check the factors. Every factor is 10, so this product is a power of ten and you can write it with an exponent.
$10 \times 10 \times 10 \times 10 \times 10$
Count the factors of 10. Point to each one as you count: 1, 2, 3, 4, 5. There are 5 factors of 10.
Write 10 once and use the count, 5, as the exponent. This works because the exponent tells how many times 10 is a factor.
$10 \times 10 \times 10 \times 10 \times 10 = 10^{5}$
Read it aloud. For an exponent of 5, use the word fifth: ten to the fifth power.

Answer

You write $10 \times 10 \times 10 \times 10 \times 10$ as $10^5$ and read it as “ten to the fifth power.”

To evaluate a power of ten means to find its value. You write that value in standard form, the usual way to write a number with digits, such as 1,000.

Start from the meaning. The exponent tells you how many factors of 10 to write. Then multiply, and let place value help:

$10^1$ is one factor of 10. That is 1 ten, or $10$ .
$10^2 = 10 \times 10$ . That is 10 tens, which make 1 hundred, or $100$ .
$10^3 = 10 \times 10 \times 10$ . That is 10 hundreds, which make 1 thousand, or $1{,}000$ .

Each new factor of 10 makes the value 10 times as large. So the digit 1 moves one place to the left, and one more 0 follows it. The chart shows this pattern.

Each new factor of 10 moves the 1 one place to the left. A dash means that place has no digit.

Power of ten	Thousands	Hundreds	Tens	Zeros after the 1
$10^1 = 10$	—	—	1	1
$10^2 = 100$	—	1	0	2
$10^3 = 1{,}000$	1	0	0	3

Key concept

The exponent counts the factors of 10

In a power of ten such as $10^3$ , the exponent tells how many times 10 is a factor. The exponent is a count. It is not a number you multiply by.

The same count tells you the zeros. $10^1$ is 10, a 1 followed by 1 zero. Each extra factor of 10 moves the 1 one more place to the left, so one more zero follows it. When the exponent is a whole number that is 1 or greater, the standard form of the power of ten is a 1 followed by zeros, and the number of zeros matches the exponent.

10^{3} = \underbrace{10 \times 10 \times 10}_{3 \text{ factors of } 10} = \underbrace{1{,}000}_{3 \text{ zeros}}

Worked example

Evaluate a power of ten

Evaluate $10^4$ . Write the value in standard form.

Read the exponent. It is 4, so 10 is a factor 4 times. Write out the four factors of 10.
$10^{4} = 10 \times 10 \times 10 \times 10$
Multiply the first two tens. That is 10 tens, which make 1 hundred. Two factors of 10 are left.
$10 \times 10 \times 10 \times 10 = 100 \times 10 \times 10$
Multiply 100 by the next 10. That is 10 hundreds, which make 1 thousand. One factor of 10 is left.
$100 \times 10 \times 10 = 1{,}000 \times 10$
Multiply 1,000 by the last 10. That is 10 thousands, which make 1 ten thousand. Now all four factors are used.
$1{,}000 \times 10 = 10{,}000$
Check with the zeros. The exponent is 4, and 10,000 is a 1 followed by 4 zeros, so the value matches the zero pattern for powers of ten.

Answer

The value of $10^4$ is $10{,}000$ .

Worked example

Go from standard form to an exponent

Write $1{,}000{,}000$ as a power of ten with an exponent. Then tell how to read it aloud.

Look at the digits. The number is a 1 followed only by zeros. Each zero after the 1 marks one place the 1 has moved to the left, and each move to the left is one factor of 10.
Count the zeros after the 1. There are 6, so 10 is a factor 6 times.
$1{,}000{,}000 = 10 \times 10 \times 10 \times 10 \times 10 \times 10$
Write 10 once with the count, 6, as the exponent, because the exponent counts the factors of 10.
$1{,}000{,}000 = 10^{6}$
Read it aloud. For an exponent of 6, use the word sixth: ten to the sixth power.

Answer

As a power of ten, $1{,}000{,}000$ is $10^6$ , and you read it as “ten to the sixth power.”

Remember: a power of ten such as $10^3$ is built from factors of 10, and the exponent counts those factors. So $10^3$ means $10 \times 10 \times 10$ . Its value is 1,000, a 1 followed by 3 zeros. The exponent is a count, not a number you multiply by.

Next, you will multiply whole numbers and decimals by powers of ten. You will see how each factor of 10 shifts every digit one place to the left.