In the table below, the number 2 is written as a factor repeatedly. The product of factors is also displayed in this table. Suppose that your teacher asked you to Write 2 as a factor one million times for homework. How long do you think that would take?
Answer : |
| Factors | Product of Factors | Description |
| 2 x 2 = | 4 | 2 is a factor 2 times |
| 2 x 2 x 2 = | 8 | 2 is a factor 3 times |
| 2 x 2 x 2 x 2 = | 16 | 2 is a factor 4 times |
| 2 x 2 x 2 x 2 x 2 = | 32 | 2 is a factor 5 times |
| 2 x 2 x 2 x 2 x 2 x 2 = | 64 | 2 is a factor 6 times |
| 2 x 2 x 2 x 2 x 2 x 2 x 2 = | 128 | 2 is a factor 7 times |
| 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = | 256 | 2 is a factor 8 times |
| Writing 2 as a factor one million times would be a very time-consuming and tedious task. A better way to approach this is to use exponents. Exponential notation is an easier way to write a number as a product of many factors. |
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The exponent tells us how many times the base is used as a factor. |
| For example, to write 2 as a factor one million times, the base is 2, and the exponent is 1,000,000. We write this number in exponential form as follows: |
| 2 | 1,000,000 | | read as two raised to the millionth power |
| Example 1: | Write 2 x 2 x 2 x 2 x 2 using exponents, then read your answer aloud. |  |
| Solution: | 2 x 2 x 2 x 2 x 2 = 25 | 2 raised to the fifth power |
Let us take another look at the table from above to see how exponents work.
Exponential
Form | Factor
Form | Standard
Form |
| 22 = | 2 x 2 = | 4 |
| 23 = | 2 x 2 x 2 = | 8 |
| 24 = | 2 x 2 x 2 x 2 = | 16 |
| 25 = | 2 x 2 x 2 x 2 x 2 = | 32 |
| 26 = | 2 x 2 x 2 x 2 x 2 x 2 = | 64 |
| 27 = | 2 x 2 x 2 x 2 x 2 x 2 x 2 = | 128 |
| 28 = | 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = | 256 |
| So far we have only examined numbers with a base of 2. Let's look at some examples of writing exponents where the base is a number other than 2. |
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| Example 2: | Write 3 x 3 x 3 x 3 using exponents, then read your answer aloud. |
| Solution: | 3 x 3 x 3 x 3 = 34 | 3 raised to the fourth power |
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| Example 3: | Write 6 x 6 x 6 x 6 x 6 using exponents, then read your answer aloud. |
| Solution: | 6 x 6 x 6 x 6 x 6 = 65 | 6 raised to the fifth power |
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| Example 4: | Write 8 x 8 x 8 x 8 x 8 x 8 x 8 using exponents, then read your answer aloud. |
| Solution: | 8 x 8 x 8 x 8 x 8 x 8 x 8 = 87 | 8 raised to the seventh power |
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| Example 5: | Write 103, 36, and 18 in factor form and in standard form. |
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| Solution: |
Exponential
Form | Factor
Form | Standard
Form |
| 103 | 10 x 10 x 10 | 1,000 |
| 36 | 3 x 3 x 3 x 3 x 3 x 3 | 729 |
| 18 | 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 | 1 |
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The following rules apply to numbers with exponents of 0, 1, 2 and 3:
| Rule | Example |
| Any number (except 0) raised to the zero power is equal to 1. | 1490 = 1 |
| Any number raised to the first power is always equal to itself. | 81 = 8 |
| If a number is raised to the second power, we say it is squared. | 32 is read as three squared |
| If a number is raised to the third power, we say it is cubed. | 43 is read as four cubed |
| Summary: | Whole numbers can be expressed in standard form, in factor form and in exponential form. Exponential notation makes it easier to write a number as a factor repeatedly. A number written in exponential form is a base raised to an exponent. The exponent tells us how many times the base is used as a factor. |
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