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rational_exponents

# rational_exponents - Rational exponents Summary of formulas...

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Rational exp onents Summary of formulas involving general integer exponents : Before considering rational exponents it is appropriate to recall definitions and properties connected with integer exponents. The following formulas define integer powers of a non-zero real number a . = a n a . a . . . . a , n factors = a ( ) - n 1 a n = a 0 1 where n is a positive integer. The following formulas in which m and n are any integers follow from the definitions. = ( ) a b n a n b n = a b n a n b n = a m a n a ( ) + m n = a m a n a ( ) - m n = ( ) a m n a ( ) m n Rational number exponents . We try to deduce a sensible definition for a r , where a is a non-negative real number and r is a general rational number, that is, = r p q , where p and q are integers and q 0. Our aim is to define a r in such a way that it retains the same meaning as before in the case that r happens to be an integer, and also such the sum rule for exponents = a r . a s a ( ) + r s still applies when r and s are general rational numbers. If the sum rule for exponents rule applies when = r s = 1 2 , then = a 1 2 . a 1 2 a + 1 2 1 2 = a ( ) 1 = a . Thus, in order for the sum rule for rational exponents to apply here, a 1 2 must be a real number with the property that, when it is multiplied with itself, it gives the value a . This means that we should identify a 1 2 with the square root of a , that is, we should make the definition = a 1 2 a . _____ In a similar way, using the sum rule, = a 1 3 . a 1 3 . a 1 3 a + + 1 3 1 3 1 3 = a ( ) 1 = a . Thus, in order for the sum rule for rational exponents to apply here, a 1 3 must be a real number with the property that, when it is multiplied with itself three times, it gives the value a . This means that we should identify a 1 3 with the cube root of a , that is, we should make the definition = a 1 3 3 a . _____

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In general, if n is a positive integer which is greater than or equal to 2, then, if the sum rule for rational exponents is to apply, we need to have: a 1 n . a 1 n . . . . . a 1 n \___________________/ n factors = a + + + 1 n 1 n . . . 1 n , with n terms in the exponent. = a ( ) 1 = a . Thus, in order for the sum rule for rational exponents to apply, a 1 n must be a real number with the property that, when it is multiplied with itself n times, it gives the value a . This means that we should define a 1 n to be the n th root of a , that is, = a 1 n n a .
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