To find the value of $x$ in this right triangle, we will use the Pythagorean theorem. The given triangle has one leg of 4, the other leg as $4\sqrt{5}$, and the hypotenuse as $16$. However, we need to use the Pythagorean theorem to find the hypotenuse $x$.

The Pythagorean theorem states: $a^2 + b^2 = c^2$ where $a$ and $b$ are the legs of the right triangle and $c$ is the hypotenuse.

Given: $a = 4, \quad b = 4\sqrt{5}$

Plugging these into the Pythagorean theorem: $4^2 + (4\sqrt{5})^2 = x^2$

Calculating the squares: $4^2 = 16$ $(4\sqrt{5})^2 = 16 \times 5 = 80$

Adding these together: $16 + 80 = 96$

So, we have: $x^2 = 96$

Taking the square root of both sides: $x = \sqrt{96}$

Simplifying the radical: $\sqrt{96} = \sqrt{16 \times 6} = \sqrt{16} \times \sqrt{6} = 4\sqrt{6}$

Therefore, the value of $x$ is: $x = 4\sqrt{6}$

Would you like any more details or have any other questions?

Here are five questions you might find helpful:

1. How do you simplify square roots?
2. What is the Pythagorean theorem and how is it applied?
3. How do you rationalize denominators in radical expressions?
4. What are some common trigonometric identities?
5. How can you solve for an unknown side in a right triangle using trigonometric ratios?

Tip: When simplifying radicals, always look for the largest perfect square factor of the number under the radical to simplify it more easily.