How can I find the value of xyz, and write my answer an a simplified radical?

x, y and z are the 3 unknew, so you need 3 equations.

In the big triangle, by using the Pythagorean's theorem:

y² + z² = (3 + 12)²

y² + z² = 15²

z² = 15² - y² ← memorize this result as (1)

In the little triangle, by using the Pythagorean's theorem:

3² + x² = y²

x² = y² - 3² ← memorize this result as (2)

In the medium triangle, by using the Pythagorean's theorem:

x² + 12² = z² → recall (1): z² = 15² - y²

x² + 12² = 15² - y² → recall (2): x² = y² - 3²

y² - 3² + 12² = 15² - y²

2y² = 15² - 12² + 3²

2y² = 90

y² = 45

→ y = 3√5

Recall (2): x² = y² - 3² = 45 - 9 = 36

→ x = 6

Recall (1): z² = 15² - y² = 15² - 45 = 180

→ z = 6√5

z^2 + y^2 = (3+12)^2

z^2 + y^2 = 225

x^2 + 3^2 = y^2

x^2 + 9 = y^2

x^2 + 12^2 = z^2

x^2 + 144 = z^2

x^2 = z^2 - 144

z^2 - 144 + 9 = y^2

z^2 + y^2 = 225

z^2 - 135 = y^2

2z^2 = 360

z^2 = 180

z = 6 root 5

180 + y^2 = 225

y^2 = 45

y = 3 root 5

x^2 = 36

x = 6

xyz = (6 x 3 x 5 x 6) = 540

I'll actually be shocked if I got this right. Honestly it's possible that geometry could have gotten me the answer quicker. But I forget geometry.