The denominator will be "rational" when each of the factors under the 4th-root radical is a 4th power of something. You need to multiply numerator and denominator by something that will make this be the case. You recognize that 8 = 2³, so to get 2⁴x⁴, you need to multiply 8x³ by 2x.
[tex]\displaystyle\frac{1}{\sqrt[4]{8x^3}}=\frac{1}{\sqrt[4]{(2x)^3}}\times\frac{\sqrt[4]{2x}}{\sqrt[4]{2x}}\\\\=\frac{\sqrt[4]{2x}}{\sqrt[4]{(2x)^3}\sqrt[4]{2x}}=\frac{\sqrt[4]{2x}}{\sqrt[4]{(2x)^4}}\\\\=\frac{\sqrt[4]{2x}}{2x}[/tex]
