We have
[tex]\sqrt{x^3} = x^{\frac32}[/tex]
so by applying the usual properties for exponents,
[tex]\left(x\sqrt{x^3}\right)^4 = \left(x \cdot x^{\frac32}\right)^4 = \left(x^{1+\frac32}\right)^4 = \left(x^{\frac52}\right)^4 = x^{\frac52\cdot4} = x^{\frac{20}2} = \boxed{x^{10}}[/tex]
