EXAMPLE 7 Simplify each expression Assume that x can represent any nonzero real number ( a ) 6 4 1 2 ( b ) ( 1 6 ) 5 4 ( c ) 6 2 5 3 4 ( d ) ( 3 2 x 5 ) 2 5 and ( e ) 1 2 5 3 2 StrategyWe will use one of the rules x m n 1 xm n or 1 x m n xm n to write the reciprocal of each exponential expression and change the exponent 's sign to positive WhyIf we can produce an equivalent expression having a positive rational exponent, we can use the methods of this section to simplify it Solution ( a ) 6 4 1 2 1 6 4 1 2 1 6 4 1 8 Because the exponent is negative, write the reciprocal of 6 4 1 2 , and change the sign of the exponent ( b ) ( 1 6 ) 5 4 is not a real number because ( 1 6 ) 5 4 is not a real number ( c ) In 6 2 5 3 4 , the base is 6 2 5 3 4 1 6 2 5 3 4 1 ( 4 6 2 5 ) 3 1 5 3 1 1 2 5 ( d ) ( 3 2 x 5 ) 2 5 1 ( 3 2 x 5 ) 2 5 1 ( 5 3 2 x 5 ) 2 1 ( 2 x ) 2 ( e ) 1 2 5 3 2 2 5 3 2 ( 2 5 ) 3 5 3 Because the exponent is negative, write the reciprocal of 1 2 5 3 2 , and change the sign of the exponent Self Check Simplify Assume that a can represent any nonzero real number ( a ) ( 3 6 ) 1 2 ( b ) ( 4 9 ) 3 2 ( c ) ( 3 2 a 5 ) 2 5 ( d ) 1 2 5 6 3 4

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EXAMPLE 7 Simplify each expression  Assume that x can represent any nonzero real number  ( a ) 6 4 1   2 ( b ) ( 1 6 ) 5   4 ( c ) 6 2 5 3   4 ( d ) ( 3 2 x 5 ) 2   5 and ( e ) 1 2 5 3   2 StrategyWe will use one of the rules x m   n   1 xm   n or 1 x m   n   xm   n to write the reciprocal of each exponential expression and change the exponent 's sign to positive  WhyIf we can produce an equivalent expression having a positive rational exponent, we can use the methods of this section to simplify it   Solution ( a ) 6 4 1   2   1 6 4 1   2   1 6 4   1 8 Because the exponent is negative, write the reciprocal of 6 4 1   2 , and change the sign of the exponent  ( b ) ( 1 6 ) 5   4 is not a real number because ( 1 6 ) 5   4 is not a real number  ( c ) In 6 2 5 3   4 , the base is   6 2 5 3   4   1 6 2 5 3   4   1 ( 4 6 2 5 ) 3   1 5 3   1 1 2 5 ( d ) ( 3 2 x 5 ) 2   5   1 ( 3 2 x 5 ) 2   5   1 ( 5 3 2 x 5 ) 2   1 ( 2 x ) 2   ( e ) 1 2 5 3   2   2 5 3   2   ( 2 5 ) 3   5 3   Because the exponent is negative, write the reciprocal of 1 2 5 3   2 , and change the sign of the exponent  Self Check Simplify  Assume that a can represent any nonzero real number  ( a ) ( 3 6 ) 1   2 ( b ) ( 4 9 ) 3   2 ( c ) ( 3 2 a 5 ) 2   5 ( d ) 1 2 5 6 3   4

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EXAMPLE 7 Simplify each expression. Assume that x can represent any nonzero real number. ( a ) 6 4 1 / 2 ( b )

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