Answer
$\left (\frac{f}{g} \right )(x)=4x\sqrt{x}$.
$(0,\infty)$.
Work Step by Step
The given functions are
$f(x)=\sqrt {48x^5}$ and $g(x)=\sqrt {3x^2}$
$\Rightarrow \left (\frac{f}{g} \right )(x)=\frac{f(x)}{g(x)}$
Substitute both functions.
$\Rightarrow \left (\frac{f}{g} \right )(x)=\frac{\sqrt {48x^5}}{\sqrt {3x^2}}$
Divide the radicands and retain the common index.
$\Rightarrow \left (\frac{f}{g} \right )(x)=\sqrt{\frac{ {48x^5}}{3x^2}}$
Divide factors in the radicand. Subtract exponents on common bases.
$\Rightarrow \left (\frac{f}{g} \right )(x)=\sqrt{16x^{5-2}}$
Simplify.
$\Rightarrow \left (\frac{f}{g} \right )(x)=\sqrt{16x^{3}}$
$\Rightarrow \left (\frac{f}{g} \right )(x)=4x\sqrt{x}$
The domain is all positive real numbers.
The interval notation is $(0,\infty)$.
