This problem shows a right triangle, where one leg is labeled as 5, the hypotenuse is labeled as 8, and the other leg is labeled as $x$, which is the value we need to solve for.

We can apply the Pythagorean theorem, which is given by:

$a^2 + b^2 = c^2$

Where:

• $a$ and $b$ are the lengths of the legs,
• $c$ is the length of the hypotenuse.

In this case:

• $a = 5$,
• $b = x$,
• $c = 8$.

Using the Pythagorean theorem:

$5^2 + x^2 = 8^2$

Let's solve for $x$.

$25 + x^2 = 64$

Subtract 25 from both sides:

$x^2 = 64 - 25$

$x^2 = 39$

Now, take the square root of both sides:

$x = \sqrt{39}$

Since $39$ is not a perfect square, this is already in its simplest radical form. Therefore, the solution is:

$x = \sqrt{39}$

Would you like more details or have any other questions? Here are some related questions to consider:

1. What are the properties of right triangles and how do they help solve problems like this?
2. Can the Pythagorean theorem be used for non-right triangles?
3. How can the square root of a number be simplified when it's not a perfect square?
4. What are some real-world applications of the Pythagorean theorem?
5. How would the equation change if the hypotenuse were unknown?

Tip: Always check if the radical can be simplified by factoring out perfect squares from the number under the square root.