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What Are Powers of 10?


A number with an exponent is called a power. Here, you will learn about powers of 10. A power of 10 is a power that has 10 as the base number (ground floor).

The exponent (attic) shows how many times 10 should be multiplied by itself.

Rule

Power of 10

10n = 10 1010 n times

10 is the base number, and n is the exponent.

Example 1

What does 104 mean?

From the power, you can see that the base number is 10 and that the exponent is 4. Therefore, 10 has to be multiplied with itself 4 times:

104 = 10 10 10 10 = 10000

What does 107 mean?

From the power, you can see that the base number is 10 and that the exponent is 7. Therefore, 10 has to be multiplied with itself 7 times:

107 = 10 10 10 10 10 10 10 = 10000000

107 = 10 10 10 10 10 10 10 = 10000000

Rule

Changing Floors

10n = 1 10n

The formula above says that you can change the floor of the power by changing the sign of the exponent. It does not matter whether you change the sign from minus to plus or from plus to minus, as long as it changes. Below, you will see some examples of how this formula works with powers of 10.

What does 107 mean? It means that the power is put in the denominator of a fraction with 1 as the numerator, while removing the minus sign in the exponent. Then, the base number 10 is multiplied by itself 7 times. It is the minus sign of the exponent that tells you to rewrite it into a fraction to make the exponent positive.

107 = 1 107 = 1 10 10 10 10 10 10 10 = 0.0000001

107 = 1 107 = 1 10 10 10 10 10 10 10 = 0.0000001

As you can see, the final number is difficult to read. To avoid losing track when working with very large or very small numbers, it’s very useful to work with powers.

Example 2

101 = 1 101 = 1 10 = 0.1 103 = 1 103 = 1 10 10 10 = 0.001 1 105 = 105 = 10 10 10 10 10 = 100000

Rule

Rules for Powers of 10

When you work with powers of 10, we have the following rules:

  • When multiplying by 10, you move the point one place to the right.

  • When multiplying by 10n, you move the point n places to the right.

  • When multiplying by 101, you move the point one place to the left.

  • When multiplying by 10n, you move the point n places to the left.

Example 3

Multiplying by Positive Powers of 10

1.3 10 = 1326.49 103 = 26490

1.3 10 = 1326.49 103 = 26490

Notice that we added a 0 when moving the point more spaces than we had digits.

Example 4

Multiplying by Negative Powers of 10

1.3 101 = 0.1326.49 103 = 0.02649

1.3 101 = 0.1326.49 103 = 0.02649

This time we also added 0’s, as we had to move the point more spaces than we had digits. As you can see, we fill the empty spaces with the 0’s we need.

If you want to practice multiplying by the power of 10, I recommend that you watch instructional videos about multiplying by the power of 10.

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