First, simplify the denominator using the rule for exponents:
x^color(red)(a) xx x^color(blue)(b) = x^(color(red)(a) +color(blue)(b))xa×xb=xa+b
(z^(1/3))/(z^(color(red)(-1/2))z^(color(blue)(1/4))) ->z13z−12z14→ z^(1/3)/z^((color(red)(-1/2)+color(blue)(1/4))z13z(−12+14) ->→ (z^(1/3))/(z^((color(red)(-2/4)+color(blue)(1/4)) z13z(−24+14) ->→
z^(1/3)/z^(-1/4)z13z−14
Next, we can simplify the fraction using another rule for exponents:
x^color(red)(a)/x^color(blue)(b) = x^(color(red)(a)-color(blue)(b))xaxb=xa−b
z^color(red)(1/3)/z^color(blue)(-1/4) = z^(color(red)(1/3)-color(blue)(-1/4)) = z^(color(red)(1/3)+color(blue)(1/4)) = z^(color(red)(4/12)+color(blue)(3/12)) = z^(color(red)(7/12)z13z−14=z13−−14=z13+14=z412+312=z712
