assuming that `x, y, z` are positive real numbers, simplify each of the following .   
 (i) `sqrt(x^(3)y^(-2))` (ii) ` (x^(-2//3) y^(-1//2))^(2)` (iii) `((x^(-4))/(y^(-10)))^((5)/(4))`   
 (iv) `(root4(81 x^(8)y^(8)z^(16))) div (root3(27x^(3)y^(6)z^(9)))`   
 (v) `root5(x^(4) root4(x^(3)root3(x^(2)sqrt(x))))`   
 (vi) `((sqrt(x))^(-(2)/(3)) sqrt(y^(4))) div (sqrt(xy^(-(1)/(2))) sqrt(x^(-2)y^(3)))`
assuming that `x, y, z` are positive real numbers, simplify each of the following .   
 (i) `sqrt(x^(3)y^(-2))` (ii) ` (x^(-2//3) y^(-1//2))^(2)` (iii) `((x^(-4))/(y^(-10)))^((5)/(4))`   
 (iv) `(root4(81 x^(8)y^(8)z^(16))) div (root3(27x^(3)y^(6)z^(9)))`   
 (v) `root5(x^(4) root4(x^(3)root3(x^(2)sqrt(x))))`   
 (vi) `((sqrt(x))^(-(2)/(3)) sqrt(y^(4))) div (sqrt(xy^(-(1)/(2))) sqrt(x^(-2)y^(3)))`




solution: (i) ` sqrt ( x ^(3) y ^( - 2 )) = sqrt ( ( x ^(3))/( y ^(2))) = (( x ^(3))/( y ^(2))) ^( (1)/(2) ) = (( x ^(3))^((1)/(2)))/(( y ^(2))^((1)/(2)) = (x ^(( 3 xx (1)/(2)))/( y ^(( 2 xx (1)/(2))) = ( x ^((3)/(2)))/( y ) `.   
 (ii) ` ( x^( - (2)/(3)) y ^( - (1)/(2)))^(2) = (x ^( - (2)/(3))^(2) ( y ^( - (1)/(3))^(2) = x ^([( - (2)/(3)) xx 2 ]) * y ^([( - (1)/(2)) xx 2 ]) `   
 ` = x ^( - (4)/(3)) * y ^( - 1 ) = (1)/( x ^((4)/(3)) y ) `.   
 (iii) ` ((x^( - 4))/( y ^( - 10 ))^((5)/(4)) = ( ( y ^( 10 ))/( x ^(4))^((5)/(4)) = (( y ^( 10 ))^((5)/(4)))/(( x ^( 4))^((5)/(4))) = ( y ^((10 xx (5)/(4)))/( x ^(( 4 xx (5)/(4))) = ( y ^((25)/(2)))/( x ^(5)) `   
 (iv) ` 4sqrt(81 x ^( 8 ) y ^( 4 ) z ^( 16)) div 3 sqrt (27 x ^( 3) y ^( 6 ) z ^(9))`   
 ` = (( 81 x ^( 8) y ^( 4 ) z ^( 16 ))^( (1)/(4)))/((27 x ^(3) y ^( 6) z^(9)) ^((1)/(3) )) = (( 81 ) ^((1)/(4)) ( x ^( 8))^((1)/(4))* ( y ^( 4))^((1)/(4))* ( z ^( 16))^((1)/(4)) )/((27)^((1)/(3))* ( x ^(3)) ^((1)/(3)) * ( y ^(6))^((1)/(3)) * ( z ^( 9 ))^((1)/(3))) `   
 ` = (( 3^(4) ) ^((1)/(4))* x^((8 xx (1)/(4)) * y ^(( 4 xx (1)/(4)) * z ^(( 16 xx (1)/(4)) )/(( 3^(3))^((1)/(3)) * x ^(( 3 xx (1)/(3)) * y ^( ( 6 xx (1)/(3)) * z ^((9 xx (1)/(3))) = ( 3^(( 4 xx (1)/(4)) * x ^(2) xx y z ^(4))/( 3 ^(( 3xx (1)/(3)) * xy ^(2) z ^(3)) `   
 ` = (x^(( 2 - 1 )) * z ^((4 -3 )))/( y ^(( 2 - 1 ))) = ( xz)/( y ) `.   
 (v) ` 5sqrt ( x ^(4) 4sqrt ( x ^(3) 3sqrt ( x ^(2) sqrt x ))) = 5 sqrt ( x ^(4) 4 sqrt ( x ^(3) 3 sqrt( x ^(2) * x ^(1//2)))) = 5sqrt( x^(4) 4 sqrt (x ^(3) 3 sqrt ( x ^((2 + (1)/(2)))))) `   
 ` = 5 sqrt ( x ^(4) 4 sqrt ( x^(3) 3 sqrt( x ^(5//2)))) = 5 sqrt( x ^(4) 4 sqrt( x^(3) * ( x ^(5//2)) ^( 1//3))) `   
 ` = 5 sqrt ( x ^( 4 ) 4 sqrt ( x ^( 3 ) * x ^( 5//6))) = 5 sqrt( x^4 4 sqrt(x^(3) * ( x ^( 5//2))^(1//3))) `   
 ` = 5 sqrt ( x ^( 4 ) 4 sqrt ( x ^( 3) * x ^(5//6)) = 5 sqrt ( x ^( 4) 4 sqrt( x ^((3 + (5)/(6)))) = 5 sqrt ( x ^(4) 4 sqrt ( x ^((23)/(6)))) `   
 ` = 5 sqrt ( x ^(4)* ( x^((23)/(6)))^((1)/(4))) = 5 sqrt( x ^(4) * x ^((23)/(24))) = 5 sqrt ( x ^(( 4 + (23)/(24))) = ( x ^((119)/(24)))^((1)/(5)) `   
 ` = x^((119)/(24) xx (1)/(5)) = x ^((119)/(120))`.   
 (vi) ` ( sqrt x) ^(- (2)/(3)) sqrt( y ^( 4)) div sqrt ( xy ^( - (1)/(2)) sqrt ( x ^( - 2) y ^(3)) `   
 ` = (( x ^( - (2)/(3)) * ( y ^(4))^((1)/(2)))/(( xy ^( - (1)/(2)))^((1)/(2)) ( x ^( - 2) y ^( 3))^((1)/(2))) = ( x^([ ((1)/(2)) xx (- (2)/(3))]) y ((4 xx (1)/(2)))/( x ^((1)/(2) * ( y ^( - (1)/(2))^((1)/(2)) *( x ^(- 2))^((1)/(2))* ( y ^(3)) ^((1)/(2))) `   
 ` = ( x ^(- (1)/(3)) * y ^(2))/( x ^((1)/(2)) * y ^([(-(1)/(2)) xx (1)/(2)]) * x ^([(-2) xx (1)/(2)] ) * y ^(( 3 xx (1)/(2))) `   
 ` = ( x ^( - (1)/(3)) * y ^(2))/( x ^((1)/(2)) * y ^( - (1)/(4)) * x ^(-1)* y ^((3)/(2))) = x ^(( - (1)/(3) - (1)/(2) + 1 )) * y ^( ( 2 + (1)/(4) - (3)/(2)) `   
 ` = x ^((1)/(6)) * y ^((3)/(4)) `. 
step by step solution by experts to help you in doubt clearance & scoring excellent marks in exams.
find answer in image to clear your doubt instantly: