5.7
Negative Exponents and Scientific Notation
Homework: EOO 1-97
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Negative Exponents:
Move the base to make the exponent a positive integer. If the base is in the
numerator with a negative exponent, move the base to the denominator and make
the exponent positive. If the base is in the denominator with a negative
exponent, move the base to the numerator and make the exponent positive.
Examples:
Rules of Exponents still
apply.
Examples:
Example:
Scientific Notation: A way of expressing very large or very small numbers that are often encountered in science.
For example, the mass of a hydrogen atom is 0.0000000000000000000000017 grams. This number written in standard (or decimal) notation is very difficult to work with in calculations, etc.
The same number, the mass of a hydrogen atom, written in scientific notation is
You may have noticed that the graphing calculator when performing a calculation that results in a very large or a very small number expresses the result in scientific notation.
Example:
The calculator allows you to write the result in scientific notation. We see that the exponent of 10 is given as 13. So 8 raised to a power of 15 is:
Converting to Scientific
Notation:
If a number in decimal notation is 10 or larger, move the decimal until the
given number is between 1 and 10, then multiply by 10 raised to a positive
exponent equal to the number of places that the
decimal was moved.
If a number in decimal notation is less than 1, move the decimal until the
given number is between 1 and 10, then multiply by 10 raised to a negative
exponent equal to the number of places that the
decimal was moved.
Example: Convert 0.00000000132 to
scientific notation.
Move the decimal until you have a number that is between 1 and 10:
This number is multiplied by a power of 10. The power or exponent is
negative since the given number was a number less than 1. The exponent is –9
since we moved the decimal nine places.
Example: Convert
1,500,000,000,000 to scientific notation.
Move the decimal until you have a number that is between 1 and 10:
This number is multiplied by a power of 10. The power or exponent is positive since the given number was a number greater than 10. The exponent is 12 since we moved the decimal twelve places.
Converting From Scientific
Notation:
If 10 has a negative exponent, move the decimal to the left the same
number of places as the exponent.
If 10 has a positive exponent, move the decimal to the
right the same number of places as the exponent.
Example: Convert from scientific notation to
decimal notation.
We must move the decimal 7 places to the left since the exponent is –7 . This means we must write 6 zeros in front of the 3.
Example: Convert from scientific
notation to decimal notation.
We must move the decimal 9 places to the right since the exponent is 9. This means we must write 9 zeros after the 2.
Multiplying or Dividing Numbers Written in Scientific
Notation:
Multiply or divide the decimal numbers as indicated. Multiply or divide
the powers of 10 using the rules of exponents for multiplication or for
division as indicated.
Example:
Example:
