Question

given all variables below are positive, express in simplest radical form with a rational denominator.
\\(\frac{\sqrt{32x^{12}}}{\sqrt{2x^2}}\\)

Explanation:

Step1: Combine the radicals

Using the property $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$, we get:
$\sqrt{\frac{32x^{12}}{2x^{2}}}$

Step2: Simplify the fraction inside the radical

Simplify the coefficients and the variables separately. For the coefficients: $\frac{32}{2} = 16$. For the variables: using the quotient rule of exponents $a^m\div a^n=a^{m - n}$, we have $x^{12}\div x^{2}=x^{12 - 2}=x^{10}$. So the expression becomes:
$\sqrt{16x^{10}}$

Step3: Simplify the square root

We know that $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ (for $a,b\geq0$) and $\sqrt{a^{2}} = a$ (for $a\geq0$). So $\sqrt{16x^{10}}=\sqrt{16}\cdot\sqrt{x^{10}}$. Since $\sqrt{16} = 4$ and $\sqrt{x^{10}}=x^{5}$ (because $(x^{5})^{2}=x^{10}$), we get:
$4x^{5}$

Answer:

$4x^{5}$