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Algebra 1

5.6.4 Using Scientific Notation

Algebra 15.6.4 Using Scientific Notation

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Activity

1. How would you write each of the following numbers in scientific notation?

Decimal Form Scientific Notation
0.00000654  
0.0000947  
683,000  
0.022  
5,200  
0.000352  

2. How would you write each of the following scientific notation values in decimal form?

Scientific Notation Decimal Form
7.26 × 10 4 7.26 × 10 4  
3.32 × 10 2 3.32 × 10 2   
9.4 × 10 6 9.4 × 10 6  
8.87 × 10 1 8.87 × 10 1  
6.395 × 10 4 6.395 × 10 4  
4.0 × 10 3 4.0 × 10 3  

Video: Converting between Scientific Notation and Decimal Form

Watch the following video to learn more about rewriting scientific notation in decimal form and decimal form in scientific notation.

Self Check

Which of the following equations is true?
  1. 2.36 × 10 5 = 0.00000236
  2. 8.5 × 10 4 = 0.00085
  3. 9.11 × 10 5 = 91 , 100
  4. 7.64 × 10 3 = 76 , 400

Additional Resources

Using Scientific Notation

Working with very large or very small numbers can be awkward. Since our number system is base ten, we can use powers of ten to rewrite very large or very small numbers to make them easier to work with. Consider the numbers 4,000 and 0.004.

Using place value, we can rewrite the numbers 4,000 and 0.004. We know that 4,000 means 4 × 1 , 000 4 × 1 , 000 and 0.004 means 4 × 1 1000 4 × 1 1000 .

If we write the 1,000 as a power of ten in exponential form, we can rewrite these numbers in this way:

4,000

4 × 1 , 000 4 × 1 , 000

4 × 10 3 4 × 10 3

0.004

4 × 1 1000 4 × 1 1000

4 × 1 10 3 4 × 1 10 3

4 × 10 3 4 × 10 3

When a number is written as a product of two numbers, where the first factor is a number greater than or equal to 1 but less than 10, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation.

A number is expressed in scientific notation when it is in this form :

a × 10 n a × 10 n , where 1 a < 10 1 a < 10 and n n  is an integer.

It is customary in scientific notation to use × as the multiplication sign, even though we avoid using this sign elsewhere in algebra.

To convert a decimal to scientific notation:

Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.

Step 2. Count the number of decimal places, n n , that the decimal point was moved.

Step 3. Write the number as a product with a power of 10. If the original number is

  • greater than 1, the power of 10 will be 10 n 10 n .
  • between 0 and 1, the power of 10 will be 10 n 10 n .

Step 4. Check.

Example

Write in scientific notation:

  1. 37,000

    Solution

    A table shows steps to convert 37,000 to scientific notation: move the decimal to get 3.7, count 4 places, so 37,000 = 3.7 times 10 to the power of 4. Steps are illustrated with text and numbers.
  2. 0.0052

    Solution

    A chart showing steps to write 0.0052 in scientific notation: moving the decimal, counting three places, writing 5.2 times 10 to the power of 3, and checking the conversion equals 0.0052.

To convert scientific notation to decimal form:

Step 1. Determine the exponent, n n , on the factor of 10.

Step 2. Move the decimal n n places, adding zeros if needed.

  • If the exponent is positive, move the decimal point n n places to the right.
  • If the exponent is negative, move the decimal point n n places to the left.

Step 3. Check.

Example

Write in decimal form:

  1. 6.2 × 10 3 6.2 × 10 3

    Solution

    A table explains 6.2 times 10 to the power of 3. The exponent is 3; move decimal right 3 places to get 6200; add zeros for placeholders; final answer is 6,200.
  2. 8.9 × 10 2 8.9 × 10 2

    Solution

    A table explaining how to write negative 8.9 times ten to the power of negative 2 in decimal form, showing the process of moving the decimal point two places left to get negative 0.089, and noting the exponent is negative 2.

Try it

Try It: Using Scientific Notation

Write in scientific notation.

1. 65,000,000

2. 0.00034

Write in decimal form.

3. 5.3 × 10 5 5.3 × 10 5

4. 4.25 × 10 3 4.25 × 10 3

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