Question

find the length of side $x$ in simplest radical form with a rational denominator.

Explanation:

Step1: Use tangent function

In a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. Here, $\theta = 30^{\circ}$, the opposite side to the $30^{\circ}$ angle is $\sqrt{7}$, and the adjacent side is $x$. So, $\tan30^{\circ}=\frac{\sqrt{7}}{x}$.
Since $\tan30^{\circ}=\frac{1}{\sqrt{3}}$, we have the equation $\frac{1}{\sqrt{3}}=\frac{\sqrt{7}}{x}$.

Step2: Cross - multiply

Cross - multiplying gives us $x = \sqrt{7}\times\sqrt{3}$.

Step3: Rationalize if needed

$x=\sqrt{21}$ is already in simplest radical form with a rational denominator (since the denominator of the overall expression for $x$ is 1).

Answer:

$\sqrt{21}$