Copyright © Cengage Learning. All rights reserved. 2 Signed Numbers and Powers of 10 Functions Copyright © Cengage Learning. All rights reserved.

Copyright © Cengage Learning. All rights reserved. 2.6 Scientific Notation Copyright © Cengage Learning. All rights reserved.

Scientific Notation Scientific Notation Scientific notation is a method that is especially useful for writing very large or very small numbers. To write a number in scientific notation, write it as a product of a number between 1 and 10 and a power of 10.

Example 1 Write 226 in scientific notation. 226 = 2.26  102 Remember that 102 is a short way of writing 10  10 = 100. Note that multiplying 2.26 by 100 gives 226.

Scientific Notation Writing a Decimal Number in Scientific Notation To write a decimal number in scientific notation, 1. Reading from left to right, place a decimal point after the first nonzero digit. 2. Place a caret (^) at the position of the original decimal point. 3. If the decimal point is to the left of the caret, the exponent of the power of 10 is the same as the number of decimal places from the caret to the decimal point.

Scientific Notation 4. If the decimal point is to the right of the caret, the exponent of the power of 10 is the same as the negative of the number of places from the caret to the decimal point. 5. If the decimal point is already after the first nonzero digit, the exponent of 10 is zero. 2.15 = 2.15  100

Example 3 Write 2738 in scientific notation.

Scientific Notation Writing a Number in Scientific Notation in Decimal Form To change a number in scientific notation to decimal form, 1. Multiply the decimal part by the given positive power of 10 by moving the decimal point to the right the same number of decimal places as indicated by the exponent of 10. Supply zeros when needed. 2. Multiply the decimal part by the given negative power of 10 by moving the decimal point to the left the same number of decimal places as indicated by the exponent of 10. Supply zeros when needed.

Example 5 Write 2.67  102 as a decimal. 2.67  102 = 267 Move the decimal point two places to the right, since the exponent of 10 is +2.

Scientific Notation You may find it useful to note that a number in scientific notation with a. a positive exponent greater than 1 is greater than 10, and b. a negative exponent is between 0 and 1. That is, a number in scientific notation with a positive exponent represents a relatively large number. A number in scientific notation with a negative exponent represents a relatively small number.

Scientific Notation Scientific notation may be used to compare two positive numbers expressed as decimals. First, write both numbers in scientific notation. The number having the greater power of 10 is the larger. If the powers of 10 are equal, compare the parts of the numbers that are between 1 and 10. Scientific notation is especially helpful for multiplying and dividing very large and very small numbers.

Scientific Notation To perform these operations, you must first know some rules for exponents. Multiplying Numbers in Scientific Notation To multiply numbers in scientific notation, multiply the decimals between 1 and 10. Then add the exponents of the powers of 10.

Example 10 Multiply (4.5  108)(5.2  10–14). Write the result in scientific notation. (4.5  108)(5.2  10–14) = (4.5)(5.2)  (108)(10–14) = 23.4  10–6 = (2.34  101)  10–6 = 2.34  10–5 Note that 23.4  10–6 is not in scientific notation, because 23.4 is not between 1 and 10.

Example 10 cont’d To find this product using a calculator that accepts numbers in scientific notation, use the following procedure. Notes: 1. You may need to set your calculator in scientific notation mode. 2. The or key is used to enter a negative number. The product is 2.34  10–5.

Scientific Notation Dividing Numbers in Scientific Notation To divide numbers in scientific notation, divide the decimals between 1 and 10. Then subtract the exponents of the powers of 10.

Example 11 Divide . Write the result in scientific notation. Using a calculator, we have The quotient is 3  104.

Scientific Notation Powers of Numbers in Scientific Notation To find the power of a number in scientific notation, find the power of the decimal between 1 and 10. Then multiply the exponent of the power of 10 by this same power.

Example 13 Find the power (4.5  106)2. Write the result in scientific notation. (4.5  106)2 = (4.5)2  (106)2 = 20.25  1012 = (2.025  101)  1012 = 2.025  1013 Note that 20.25 is not between 1 and 10.

Example 13 cont’d The result is 2.025  1013.