ANSWER
[tex] {2x}^{\frac{8}{3} } - 5 {x}^{ \frac{5}{3} } + \frac{ 4 }{ {x}^{ \frac{1}{3} } } [/tex]
EXPLANATION
The given rational expression is
[tex] \frac{2 {x}^{3} - 5 {x}^{2} + 4 }{ {x}^{ \frac{1}{3} } } [/tex]
We share the denominator for each numerator to obtain,
[tex] \frac{2 {x}^{3} }{ {x}^{ \frac{1}{3} } } + \frac{ - 5 {x}^{2} }{ {x}^{ \frac{1}{3} } } + \frac{ 4 }{ {x}^{ \frac{1}{3} } } [/tex]
This is the same as
[tex] \frac{2 {x}^{3} }{ {x}^{ \frac{1}{3} } } - \frac{ 5 {x}^{2} }{ {x}^{ \frac{1}{3} } } + \frac{ 4 }{ {x}^{ \frac{1}{3} } } [/tex]
Recall that,
[tex] \frac{ {a}^{m} }{ {a}^{n} } = {a}^{m - n} [/tex]
We apply this property to obtain;
[tex] {2x}^{3 - \frac{1}{3} } - 5 {x}^{2 - \frac{1}{3} } + \frac{ 4 }{ {x}^{ \frac{1}{3} } } [/tex]
We now simplify the exponents to get;
[tex] {2x}^{\frac{8}{3} } - 5 {x}^{ \frac{5}{3} } + \frac{ 4 }{ {x}^{ \frac{1}{3} } } [/tex]
