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simplify (a^(1/2)b)^(1/2)(ab^(1/2))

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Its a little difficult deciphering your notation, but assuming the expression is (a1/2b)1/2ab1/2, you can simplify as follows:
First distribute the exponent to the upper right of the first parenthesis (1/2), by raising each term inside to the power shown of 1/2.  Note that when raising a variable with an existing exponent to another exponent, the exponents should be multiplied.  So a1/2 raised to 1/2 equals a 1/2*1/2 or a1/4.  The resulting expression is as follows:  a1/4b1/2ab1/2
Next reorder to bring "like" variables together:  a1/4ab1/2b1/2
Finally complete the multiplication of "like" variables by adding the exponents:  a(1/4+1)b(1/2+1/2)
Since 1/4 + 1=4/4+1/4=5/4 and 1/2+1/2=1, the final result is a5/4b
Hope this helps!!