5.6 Rules of Exponents and Scientific Notation 267 Consider (2 ) . 3 2 = = = + (2 ) (2 )(2 ) 2 2 3 2 3 3 3 3 6 This example illustrates the power rule for exponents. Negative Exponent Rule = ≠ −a a a 1 , 0 m m Example 4 Using the Negative Exponent Rule Use the negative exponent rule to simplify. a) −3 2 b) −7 1 Solution a) = = −3 1 3 1 9 2 2 b) = = −7 1 7 1 7 1 1 Now try Exercise 15 Did You Know? A Very Large Number The number, 170,141,183,460, 469,231,731,687,303,715,884, 105,727 is a very large number. How would you read this number? Take a breath. 170 undecillion, 141 decillion, 183 nonillion, 460 octillion, 469 septillion, 231 sextillion, 731 quintillion, 687 quadrillion, 303 trillion, 715 billion, 884 million, 105 thousand, 727. Often numbers this large can be represented with an approximation involving scientific notation. This number, however, is a prime number, and its exact representation is very important. Another exact, and more efficient, representation of this number is 2 1. 127 − Power Rule for Exponents = ⋅ a a ( )m n m n Thus, = = ⋅ (2 ) 2 2 . 3 2 3 2 6 Example 5 Using the Power Rule for Exponents Use the power rule for exponents to simplify. a) (2 )4 5 b) (3 )7 2 Solution a) = = ⋅ (2 ) 2 2 4 5 4 5 20 b) = = ⋅ (3 ) 3 3 7 2 7 2 14 Now try Exercise 19 Summary of the Rules of Exponents ⋅ = + a a a m n m n Product rule for exponents = =/ − a a a a , 0 m n m n Quotient rule for exponents = =/ a a 1, 0 0 Zero exponent rule = =/ −a a a 1 , 0 m m Negative exponent rule = ⋅ a a ( )m n m n Power rule for exponents 7 7
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