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Scientific Notation Scientific Notation A number is expressed in scientific notation when it is in the form a x 10n where a is between 1 and 10 and n is an integer An ordinary penny contains about 20,000,000,000,000,000,000,000 atoms. The average size of an atom is about 0.00000003 centimeters across. The length of these numbers in standard notation makes them awkward to work with. Scientific notation is a shorthand way of writing such numbers. In scientific notation the number of atoms in a penny is 2.0 1022, and the size of each atom is 3.0 10–8 centimeters across. Helpful Hint The sign of the exponent tells which direction to move the decimal. A positive exponent means move the decimal to the right, and a negative exponent means move the decimal to the left. Write the width of the universe in scientific notation. 210,000,000,000,000,000,000,000 miles Where is the decimal point now? After the last zero. Where would you put the decimal to make this number be between 1 and 10? Between the 2 and the 1 2.10,000,000,000,000,000,000,000. How many decimal places did you move the decimal? 23 When the original number is more than 1, the exponent is positive. The answer in scientific notation is 2.1 x 1023 1) Express 0.0000000902 in scientific notation. Where would the decimal go to make the number be between 1 and 10? 9.02 The decimal was moved how many places? 8 When the original number is less than 1, the exponent is negative. 9.02 x 10-8 Write 28750.9 in scientific notation. 1. 2. 3. 4. 2.87509 x 10-5 2.87509 x 10-4 2.87509 x 104 2.87509 x 105 Additional Example 1A: Translating Scientific Notation to Standard Notation Write the number in standard notation. A. 1.35 105 1.35 10 5 105= 100,000 1.35 100,000 135,000 Since the exponent is a positive 5 move to the right 5 spaces. You move to the right when the exponent is positive because the number must be greater than 1 when the exponent is positive. Additional Example 1B: Translating Scientific Notation to Standard Notation Continued Write the number in standard notation. B. 2.7 10 –3 2.7 10 –3 10 –3 1 2.7 100 100 2.7 0.0027 = 1 100 Divide by the reciprocal. Think: Move the decimal left 3 places. Move to the left to make the number less than one since the exponent is negative, move three places because the exponent is 3. Lesson Quiz Write in standard notation. 1. 1.72 104 17,200 2. 6.9 10–3 0.0069 Write in scientific notation. 3. 0.0053 5.3 10–3 4. 57,000,000 5.7 107 5. A human body contains about 5.6 x 106 microliters of blood. Write this number in standard notation. 5,600,000 Write in PROPER scientific notation. (Notice the number is not between 1 and 10) 8) 234.6 x 109 2.346 10 92 Move the decimal two places. Since the number was greater than 1 add 2 to the exponent 2.346 10 11 9) 0.0642 x 104 6.42 10 42 6.42 10 2 Move the decimal two places. Since the number is less than 1 subtract 2 from the exponent Write 531.42 x 105 in scientific notation. 1. 2. 3. 4. 5. 6. 7. .53142 x 102 5.3142 x 103 53.142 x 104 531.42 x 105 53.142 x 106 5.3142 x 107 .53142 x 108 Multiplying with Scientific Notation (2.3 X 102)(3.3 X 103) • Multiply the Coefficients • 2.3 X 3.3 = 7.59 • Add the Exponents • 102 X 103 = 105 • 7.59 X 105 • 759,000 Multiplying with Scientific Notation • (4.6 X 104) X (5.5 X 103) = ? 25.3 10 7 2.53 10 8 This is not in proper scientific notation, so get in in the correct form. Remember since you moved one place and the number was greater than 1 add it to the exponent. • (3.1 X 103) X (4.2 X 105) = ? 13.02 10 9 1.302 10 8 This is not in proper scientific notation, so get in in the correct form. Remember since you moved one place and the number was greater than 1 add it to the exponent. Dividing with Scientific Notation • (3.3 X 104)/ (2.3 X 102) • Divide the Coefficients • 3.3/ 2.3 = 1.434783 • Subtract the Exponents • 104 / 102 = 102 • 1.4347823 X 102 • 143.4783 Dividing with Scientific Notation 4.6 10 2 5.5 10 4 .8363636364 10 2 8.3636364 10 1 3.110 3 4.2 10 5 .7380952381102 7.380952381103 Since the answer is not in proper form, you must get it in proper form. The decimal must be moved one place. Since the number is less than one, you must subtract 1 from the exponent 2 Since the answer is not in proper form, you must get it in proper form. The decimal must be moved one place. Since the number is less than one, you must subtract 1 from the exponent -2 5) Use a calculator to evaluate: 7.2 x 10-9 1.2 x 102 On the calculator, the answer is: 6 E -11 The answer in scientific notation is 6 x 10 -11 The answer in standard notation is 0.00000000006 7) Use a calculator to evaluate (3,600,000,000)(23). On the calculator, the answer is: 8.28 E 10 The answer in scientific notation is 8.28 x 10 10 The answer in standard notation is 82,800,000,000 Write (2.8 x 103)(5.1 x 10-7) in scientific notation. 1. 2. 3. 4. 14.28 x 10-4 1.428 x 10-3 14.28 x 1010 1.428 x 1011