integer_exponents

integer_exponents - Integer Exponents A real number raised...

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Integer Exponents A real number raised to a positive integer power . We start with a few examples involving powers of 7. = 7 2 7 x = 7 49 = 7 3 7 x 7 x = 7 49 x = 7 343 = 7 4 7 x 7 x 7 x = 7 343 x = 7 2401. In general, if a is a real number, and n is a positive integer, then = a n a . a . a . . . . a ------- (i) \_____________/ n times __________ The value of a n is obtained by multiplying a by itself n times. Here are some more examples. = ( ) - 3 4 ( ) - 3 . ( ) - 3 . ( ) - 3 . = ( ) - 3 81 = ( ) - 3 5 ( ) - 3 . ( ) - 3 . ( ) - 3 . ( ) - 3 . = ( ) - 3 - 243 = - 3 4 - ( ) 3 . 3 . 3 . 3 = - 81 = 3 2 3 3 2 . 3 2 . = 3 2 27 8 = - 2 5 6 - 2 5 . - 2 5 . - 2 5 . - 2 5 . - 2 5 . = - 2 5 ( ) - 2 6 5 6 = 2 6 5 6 = 64 15625 In general, we have the formula = a b n a n b n ------- (ii), ______ where a and b are real numbers with b 0, and n is a positive integer. For example, = π 2 4 π 4 16 = 2 3 4 2 . 2 . 2 . 2 3 4 = ( ) 2 . 2 . ( ) 2 . 2 3 4 = 2 . 2 3 4 = 4 81 . Similarly, = ( ) a b n a n b n ------- (iii), ______ where a and b are real numbers and n is a positive integer. For example, = ( ) 2 . 3 4 ( ) 2 . 3 . ( ) 2 . 3 . ( ) 2 . 3 . ( ) 2 . 3 = ( ) 2 . 2 . 2 . 2 . ( ) 3 . 3 . 3 . 3 = 2 4 . 3 4 = 16 x = 81 1296.
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Evaluating the product 2 . 3 inside the bracket first gives the same result: ( ) 2 . 3 4 = = 6 4 1296. In general, = ( ) a b n ( ) a b x ( ) a b x . . . . x ( ) a b \_______________________/ n times = ( ) a . a . . . a x ( ) b . b . . . b \_________/ \_________/ n times n times = a n x b n . For example, = ( ) 2 π 4 2 4 . = π 4 16 π 4 . Also = ( ) 5 3 3 5 3 . = 3 3 125 . 3 . 3 . 3 = 125 . 3 . = 3 375 3 If m and n are positive integers, then a m . = a n a ( ) + m n ------- (iv) _______ For example, a 4 . a 3 = ( ) a . a . a . a x ( ) a . a . a = a . a . a . a . a . a . a = a 7 . In general, a m . = a n ( ) a . a . . . a x ( ) a . a . . . a \_________/ \_________/ m times n times = ( ) a . a . a . . . a \____________/ + m n times = a ( ) + m n . For example, 3 4 x = 3 2 3 6 = 729. If m > n and a 0, then = a m a n a ( ) - m n ------- (v). ______ For example, = a 5 a 3 a . a . a . a . a a . a . a = a . a . a a . a . a . a . a = 1 . a . a = a 2 .
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In general, a m a n = a n . a ( ) - m n a n = a n a n . a ( ) - m n = 1 . a ( ) - m n = a ( ) - m n . For example, = 10 11 10 8 10 3 = 1000. If m < n and a 0, then = a m a n 1 a ( ) - n m ------- (vi). ______ For example,
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This note was uploaded on 11/12/2011 for the course MATH 111 taught by Professor Uri during the Spring '08 term at Rutgers.

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integer_exponents - Integer Exponents A real number raised...

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