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Exponents & Scientific Notation Introductory Algebra 1 Exponents & Scientific Notation Integer Exponents Scientific Notation Power Rules 2 Integral Exponents & Scientific Notation 3 Positive Integral Exponents 4 Negative Integral Exponents Rules n n n 1 1 a b 1 n 1 n , a , a , and n a a a b a a 5 Negative Integral Exponents If a is any nonzero real number and n is a positive integer, then a-n = 1/an. (Note that a negative exponent is not the same or related to a negative number.) 7p 4 1 7 7 4 4 p p 3 3 1 9 6 Negative Integral Exponents If a is any nonzero real number and n is a positive integer, then a-n = 1/an. (Note that a negative exponent is not the same or related to a negative number.) 1 3 3 3 1 1 3 3 1 1 3 9 9 9 1 9 1 32 1 2 3 1 3 3 3 3 3 1 3 1 3 7 Negative Integral Exponents If a is any nonzero real number and n is a positive integer, then a-n = 1/an. (Note that a negative exponent is not the same or related to a negative number.) 3 3 3 3 1 3( 2 ) 1 or 1 2 3 3 9 3 3 8 Negative Integral Exponents If a is any nonzero real number and n is a positive integer, then a-n = 1/an. (Note that a negative exponent is not the same or related to a negative number.) 3 1 3 9 9 1 or 1 3 9 3 27 3 9 Product Rule for Exponents a a a m n mn e.g. 2 2 2 2 2 2 2 2 2 3 2 3 2 5 (note that the bases must be the same.) 3x y 5xy ? 2 3 4 10 Product Rule for Exponents a a a m n mn e.g. 2 2 2 2 2 2 2 2 2 3 2 3 2 5 (note that the bases must be the same.) 3x y 5xy 3 5 x x y y 2 3 4 2 3 4 11 Product Rule for Exponents a a a m n mn e.g. 2 2 2 2 2 2 2 2 2 3 2 3 2 5 (note that the bases must be the same.) 2 3 4 2 3 4 3 x y 5 xy 3 5 x x y y 15 x 2 1 15 x y 3 y 7 3 4 12 Zero Exponents 13 Zero Exponents 3 2 3 3 0 e.g. 1 3 2 2 2 14 Changing the Sign of an Exponent an and a-n are reciprocals, therefore a-n = (1/an) and (1/a-n) = an 3 3a 1 1 1 3 3 4 21a a 21 1 4 a 3 1 4 a 3 a 7 21 a 15 2 2 22 e.g . 3 223 21 2 2 2 2 am mn a an (note that the bases must be the same.) 2 3 3 x y 5xy 4 2 3 3 x y 5 x y 4 3 21 3 4 x y 5 3 1 1 x y 5 16 am mn a an 2 2 22 e.g . 3 223 21 2 2 2 2 (note that the bases must be the same.) 3x y 3 x y 5xy 5 x y 2 3 4 2 3 4 3 21 3 4 x y 5 3 1 1 3x x y 5 5y 17 Scientific Notation Format: a × 10n × used for multiplication n an integer 1 a 10 Example: 2 × 103 = 2,000 18 Scientific Notation Examples 5,280 = ? .14159 = ? 19 Scientific Notation Examples 5,280 = 5.28 × 103 .14159 = 1.4159 × 10-1 20 Converting Standard to Scientific 1. Count the number of places (n) that the decimal point must be moved so that it will follow the first nonzero digit of the number. 2. If the original number was larger than 10, use 10n. 3. If the original number was smaller than 1, use 10-n. 21 Converting Scientific to Standard 1. Determine the number of places to move the decimal point by examining the exponent on the 10. 2. Move to the right for a positive exponent and to the left for a negative exponent. 22 Multiplication Example (2,000,000) (0.0000000008) ( 2 × 106 ) ( 8 × 10-10 ) = (2)(8)(106)(10-10) = 16 × 10-4 = 1.6 × 10-3 = 0.0016 Scientific Notation is designed for multiplication and division, not addition and subtraction. 23 Light from the Sun Distance to the sun ~ 93,000,000 miles Speed of light ~ 186,000 miles per second Find how long it takes for the light from the sun to reach the earth. 24 Light from the Sun Distance to the sun ~ 93,000,000 miles Speed of light ~ 186,000 miles per second Find how long it takes for the light from the sun to reach the earth. 93,000,000 9.310 7 186,000 1.86 10 5 25 Light from the Sun Distance to the sun ~ 93,000,000 miles Speed of light ~ 186,000 miles per second Find how long it takes for the light from the sun to reach the earth. 93,000,000 9.3 10 7 186,000 1.86 10 5 9.3 107 9.3 7 5 2 10 5 10 5 1.86 10 1.86 26 Light from the Sun Distance to the sun ~ 93,000,000 miles Speed of light ~ 186,000 miles per second Find how long it takes for the light from the sun to reach the earth. 93,000,000 9.3 10 7 186,000 1.86 10 5 9.3 107 9.3 7 5 2 10 5 10 500 sec 5 1.86 10 1.86 27 Warm-ups 28 The Power Rules 29 Power of a Power Rule 2 2 2 2 2 3 2 2 2 2 22 22 2 26 2 23 30 Power of a Power Rule x x 2 1 3 3 31 Power of a Power Rule x x 2 1 3 3 2 x 1 2 9 11 9 x x 11 x x 32 Power of a Product 2x 2 x 3 4 4 3 4 16 x 12 33 Power of a Product 3x 3 y 2 4 34 Power of a Product 3x 3 y 2 4 3 4 x y 3 4 1 12 8 x y 81 12 x 8 81y 2 4 35 Power of a Quotient 3 x x x x x x 3 27 3 3 3 3 3 3 3 3 3 x 3 3 3 3 27 3 x x x x 3 x 36 Power of a Quotient 2 3 3 4x 37 Power of a Quotient 3 3 4x 2 4x 3 3 2 4x 2 3 4 x 3 2 2 9 3 2 16 x 9 38 6 Variable Exponents 2 m 3 n 5n 39 Variable Exponents 2 2 m 5n m 3 3 n 5n n 5n n5 n 5n2 2 2 m5n 5mn 3 3 40 Integral Exponents 41 Warm-ups 42 Next Slide Show 43