In some fields, we need to work with very large or very small numbers. It can be inconvenient to write these numbers in decimal form, so we can use scientific notation to express these numbers in a shorter form and make them easier to work with. For example, the speed of light is approximately 300 000 000 m/s, but can be expressed in scientific notation as 3.0 times 10 to the power of 8 m/s. This module will show you how to convert between decimal notation and scientific notation, and how to multiply and divide in scientific notation.
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Write 0.0000205 in scientific notation.
Line 1. Move the decimal to the right of the first non-zero digit.
Line 2. Count the number of decimal places, n, that the decimal point was moved. The decimal point was moved 5 places to the right, so the number is 2.05.
Line 3. Write the number as a product with a power of 10. Since the decimal was moved to the right, n is negative and the number written in scientific notation is 2.05 times 10 to the negative 5.
Example 2. Write 1.2 times 10 to the power of 9 in decimal form.
Line 1. Determine the exponent on the factor of 10 is 9. Since the exponent is positive move the decimal 9 places to the right.
Line 2. The exponent is positive, so move the decimal point 9 places to the right adding zeros as needed. The number in decimal form is 1 billion 200 million, or 1 200 000 000.
Multiply open parenthesis 2.0 times 10 to the negative 6 close parenthesis open parenthesis 3.2 times 10 to the negative 7. Keep your answer in scientific notation.
Line 1. Rearrange the factors. 2.0 times 3.2 times 10 to the negative 6 times 10 to the negative 7.
Line 2. Multiply 2.0 by 3.2 and use the Product Property to multiply 10 to the negative 6 and 10 to the negative 7, so the expression is 6.4 times 10 to the negative 6 plus negative 7 end superscript.
Line 3: Add the exponents to simplify to the final answer 6.4 times 10 to the negative 13.
AODA: Divide. 1.2 times 10 to the 8 over 4.8 times 10 to the negative 5. Step 1. Separate the factors and rewrite the expression as 1.2 over 4.8 times 10 to the 8 over 10 to the negative 5.
Step 2. Divide 1.2 by 4.8 and use the Quotient Property to divide 10 to the 8 and 10 to the negative 5, so the expression is 0.25 times 8 to the 8 minus negative 7 end superscript.
Step 3. Simplify the exponents so the expression is 0.25 times 10 the 13.
Step 4: Notice that this is not in proper scientific notation because 0.25 is less than 1, so rewrite 0.25 in scientific notation, so the expression is open parenthesis 2.5 times 10 to the negative 1 close parenthesis times 10 to the 13.
Step 5. Simplify using the Product Property to multiply 10 to the negative 1 and 10 to the 13, so the final answer in scientific notation is 2.5 times 10 to the 12.
Examples Source: "Prealgebra - opens in a new window" by Lynn Marecek & Mary Anne Anthony-Smith is licensed under CC BY 4.0 - opens in a new window / A derivative from the original work - opens in a new window