a x 10n, Scientific Notation Scientific Notation A number written in the form a x 10n, where n is an integer and a is greater than or equal to 1 and less than 10, is said to be in scientific notation.

Writing a Number in Scientific Notation Step 1 Move the decimal point to the right of the first nonzero digit. Step 2 Count the number of places you moved the decimal point. The number of places is the absolute value of the exponent on 10. Step 3 If the original number was “large” (10 or more), the exponent on 10 is positive. If the number was “small” (between 0 and 1), the exponent is negative.

Writing Numbers in Scientific Notation EXAMPLE 1 Writing Numbers in Scientific Notation Write each number in scientific notation. ( a ) 470,000,000 = 4.7 x 108 = 4.7 x 10n Step 1 Move the decimal point to the right of the first nonzero digit. Step 2 Count the number of places you moved the decimal point. Step 3 The original number was “large” so the exponent is positive. . 4 7 0 0 0 0 0 0 0 Move the decimal point 8 places.

Writing Numbers in Scientific Notation EXAMPLE 1 Writing Numbers in Scientific Notation Write each number in scientific notation. ( b ) 0.000032 = 3.2 x 10n = 3.2 x 10– 5 Step 1 Move the decimal point to the right of the first nonzero digit. Step 2 Count the number of places you moved the decimal point. Step 3 The original number was “small” so the exponent is negative. . 0 . 0 0 0 0 3 2 Move the decimal point 5 places.

Writing Numbers in Scientific Notation EXAMPLE 1 Writing Numbers in Scientific Notation Write each number in scientific notation. ( c ) 0.000000000829 = 8.29 x 10n = 8.29 x 10– 10 9 zeroes Step 1 Move the decimal point to the right of the first nonzero digit. Step 2 Count the number of places you moved the decimal point. Step 3 The original number was “small” so the exponent is negative. . 0 . 0 0 0 0 0 0 0 0 0 8 2 9 Move the decimal point 10 places.

Writing Numbers in Scientific Notation EXAMPLE 1 Writing Numbers in Scientific Notation Write each number in scientific notation. ( d ) 5,100,000,000,000,000,000 = 5.1 x 10n = 5.1 x 1018 Step 1 Move the decimal point to the right of the first nonzero digit. Step 2 Count the number of places you moved the decimal point. Step 3 The original number was “large” so the exponent is positive. . 5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Move the decimal point 18 places.

Converting Scientific Numbers to Numbers without Exponents To convert a number written in scientific notation to a number without exponents, remember that multiplying by a positive power of 10 will make the number larger; multiplying by a negative power of 10 will make the number smaller.

Writing Numbers without Exponents EXAMPLE 2 Writing Numbers without Exponents Write each number without exponents. ( a ) 9.2 x 104 Since the exponent is positive, make 9.2 larger by moving the decimal point 4 places to the right, inserting zeros as needed. 9.2 x 104 = 9.2000 = 92,000

Writing Numbers without Exponents EXAMPLE 2 Writing Numbers without Exponents Write each number without exponents. ( b ) 5.38 x 107 Since the exponent is positive, make 5.38 larger by moving the decimal point 7 places to the right, inserting zeros as needed. 5.38 x 107 = 5.3800000 = 53,800,000

Writing Numbers without Exponents EXAMPLE 2 Writing Numbers without Exponents Write each number without exponents. ( c ) 6.17 x 10– 5 Move 5 places to the left; multiplying by a negative power of 10 makes the number smaller. 6.17 x 10– 5 = 00006.17 = 0.0000617

The Exponent NOTE As shown in Example 2, the exponent tells the number of places and the direction that the decimal point is moved. Positive exponents move the decimal point to the right. 5.38 x 107 = 53,800,000 Negative exponents move the decimal point to the left. 6.17 x 10– 5 = 0.0000617

Multiplying and Dividing with Scientific Notation EXAMPLE 3 Multiplying and Dividing with Scientific Notation Write each product or quotient without exponents. ( a ) ( 5 x 108 ) ( 7 x 10– 2 ) = ( 5 x 7 ) ( 108 x 10– 2 ) Commutative & associative properties = 35 x 106 Product rule for exponents = 35,000,000 Write without exponents.

Multiplying and Dividing with Scientific Notation EXAMPLE 3 Multiplying and Dividing with Scientific Notation Write each product or quotient without exponents. ( b ) ( 8 x 10 – 6 ) ( 4 x 10 2 ) = 8 4 10– 6 10 2 x = 2 x 10– 8 = 0.00000002

Solving an Application Problem EXAMPLE 4 Solving an Application Problem Pluto is approximately 6 x 109 miles from the sun. The speed of light is approximately 3 x 105 km/s. How many hours does it take for light to travel from the sun to Pluto? ( 6 x 109 ) ( 3 x 105 ) = 6 3 109 105 x = 2 x 104 = 20,000 hours ≈ 2.3 years!

Solving an Application Problem EXAMPLE 5 Solving an Application Problem A standard piece of paper is approximately 0.0038 inches. How thick would 1,000 pieces of paper be? ( 3.8 x 10– 3 ) ( 1 x 103 ) = ( 3.8 x 1 ) ( 10– 3 x 103 ) = 3.8 x 100 = 3.8 x 1 = 3.8 inches

Solving an Application Problem EXAMPLE 6 Solving an Application Problem In 2008, Warren Buffet was worth about $62,000,000,000. If he decided to give all of his money away, how many millionaires could he make? ( 6 .2 x 1010 ) ( 1 x 106 ) = 6.2 1 1010 106 x = 6.2 x 104 = 62,000 millionaires