Question

given all variables below are positive, express in simplest radical form with a rational denominator.\\(\frac{\sqrt{14x^8}}{\sqrt{7x^2}}\\)

Explanation:

Step1: Combine the radicals

Using the property $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$, we can combine the two square roots into one:
$\frac{\sqrt{14x^{8}}}{\sqrt{7x^{2}}}=\sqrt{\frac{14x^{8}}{7x^{2}}}$

Step2: Simplify the fraction inside the radical

Simplify the coefficients and the variables separately. For the coefficients, $\frac{14}{7} = 2$. For the variables, using the quotient rule of exponents $a^{m}\div a^{n}=a^{m - n}$, we have $x^{8}\div x^{2}=x^{8 - 2}=x^{6}$. So the expression inside the radical becomes:
$\sqrt{2x^{6}}$

Step3: Simplify the square root

We can rewrite $x^{6}$ as $(x^{3})^{2}$ since $(x^{3})^{2}=x^{3\times2}=x^{6}$. Then, using the property $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ (where $a = 2$ and $b=(x^{3})^{2}$) and $\sqrt{(x^{3})^{2}}=x^{3}$ (since $x$ is positive), we get:
$\sqrt{2}\cdot\sqrt{(x^{3})^{2}}=\sqrt{2}x^{3}$

Answer:

$x^{3}\sqrt{2}$