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Scientific Notation

Converting From Expanded Form to Scientific Notation

Here’s how to convert a number from expanded form to scientific notation.

 

Procedure — To Convert From Expanded Form to Scientific Notation

Step 1 Move the decimal point until there is only one nonzero digit to its left.

Step 2 Write the number from Step 1 multiplied by a power of 10. To find the power, count the number of places you moved the decimal point.

• If you moved the decimal point to the left, the power is positive.

• If you moved the decimal point to the right, the power is negative.

Step 3 Check the sign of the power.

• If the original number was greater than or equal to 10, the power should be positive.

• If the original number was greater than 0 and less than 1, the power should be negative.

 

Example 1

The distance from the sun to the earth is approximately 93,000,000 miles.

Write 93,000,000 in scientific notation.

Solution

Step 1 Move the decimal point until there is only one nonzero digit to its left.    
The original number is not written with a decimal point.   93,000,000
Since the number is a whole number, we can rewrite it with a decimal point to the right of the last digit.   = 93,000,000.
Then we move that decimal point seven places to the left.  
Step 2 Write the number from Step 1 multiplied by a power of 10.

We moved the decimal point 7 places to the left.

So the power of 10 is 7.

  = 9.3 × 107

Step 3 Check the sign of the power.

The original number is greater than 10, so the power should be positive, and it is.

We can also check our result by expanding 9.3 × 107.

9.3 × 107

= 9.3 · 10 · 10 · 10 · 10 · 10 · 10 · 10

= 93,000,000

Thus, 93,000,000 miles = 9.3 × 107 miles.

 

Example 2

The rest mass of an electron is approximately 0.000000000000000000000000000000911 kg.

Write this number in scientific notation.

Solution

Step 1 Move the decimal point until there is only one nonzero digit to its left.

Move the decimal point until it is between the 9 and the 1.

Step 2 Write the number from Step 1 multiplied by a power of 10.

We moved the decimal point 31 places to the right.

So the power of 10 is -31.

 

Step 3 Check the sign of the power. The original number is less than1, so the power should be negative, and it is.

Thus, 0.000000000000000000000000000000911 kg = 9.11 × 10-31 kg.