Simplify $\dfrac{(2x^2)^3}{4\sqrt{x}}$, giving your answer in the form of $ax^n$, where $a$ and $n$ are rational numbers. Answer ...

1. Expand the power in the numerator using laws of indices.

$(2x^2)^3=2^3(x^2)^3=8x^6$
So
$\frac{(2x^2)^3}{4\sqrt{x}}=\frac{8x^6}{4\sqrt{x}}.$

2. Simplify the coefficient and subtract indices for the powers of $x$.

Since $\sqrt{x}=x^{\frac12}$,
$\frac{8x^6}{4\sqrt{x}}=\frac{8x^6}{4x^{\frac12}}=2x^{6-\frac12}=2x^{\frac{11}{2}}.$