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When the coefficient is not between 1 and 10, numbers can be adjusted to scientific notation.
To do this:
1) Move the decimal, right or left, so the first number is between 1 and 10, then compensate by
2) Increasing or decreasing the power of 10 by as many as the number of places you moved the decimal
Note: We are not changing the overall number.
So when one part of the number goes up, the other goes down and vice versa.
One part of the number is compensating for the other part to keep the overall number equal.
Example 1: Adjust 45.891 x 107so that it's in scientific notation.
Step 1:
We have to move the decimal in 45.891 one place to the left in order to make it between 1 and 10:
which decreases the # by one power of 10 to 4.5891
Step 2: To compensate for this decrease by one, we increase the power of 10 by one: 107 → 108
Example 2: Adjust .000712 x 10-6 so that it's in scientific notation.
| Step 1: We have to move the decimal in .000712 four places to the right in order to make it between 1 and 10: |
.000712  7.12 which increases number by 4 powers of 10 to 7.12 |
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| Moved decimal four to right |
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Step 2: To compensate for this increase by four powers of 10, we have to decrease the power of 10 by four.
(Reminder: 4 less than -6 is -10.) |
| 10-6 10-10 |
| This number got bigger by 4 decimals | |
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| So this number got smaller by 4 powers | |
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| So: .000712 x 10-6 = 7.12 x 10-10 |
| adjusting the number into scientific notation | |
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| Remember: -10 is smaller than -6 |
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Example 3: Coefficient Goes Up as Exponent Goes Down so Number Stays the Same. There are ways you can write 3,960,000 as a number times a power of 10 up to a power of 6. As you know, as you move down, the coefficient increases as the power of 10 decreases. This is because...
...the two numbers compensate for each other and the overall number stays the same.
overall number
stays the same
because... |
| ...1st number
goes down as... |
| ...2nd number goes
up until... |
|  | | | | 3,960,000 = | 3,960,000 | * 100 | | 3,960,000 = | 396,000 | * 101 | | 3,960,000 = | 39,600 | * 102 | | 3,960,000 = | 3,960 | * 103 | | 3,960,000 = | 396 | * 104 | | 3,960,000 = | 39.6 | * 105 | | 3,960,000 = | 3.96 | * 106 |
| | |  | | | | ...this number is now between 1 and 10 |
| | |  |  | ...we arrive at
scientific notation
because... |
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The final answer in scientific notation is 3968000 = 3.96 * 10 6 as it is in between 1 to 10.
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