1. Given a right triangle with angles 60\degree, 90\degree, and the sides labeled as follows: side opposite 60\degree angle = $x$, side adjacent to 60\degree angle = 6, and hypotenuse = 15. 2. We know from trigonometry that for the 60\degree angle: $$\tan 60^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{x}{6}$$ 3. Using the exact value $\tan 60^\circ = \sqrt{3}$, we have: $$\sqrt{3} = \frac{x}{6}$$ 4. Solve for $x$ by multiplying both sides by 6: $$x = 6 \sqrt{3}$$ 5. Double check using the Pythagorean theorem: hypotenuse squared equals sum of squares of legs $$15^2 = x^2 + 6^2 \Rightarrow 225 = x^2 + 36 \Rightarrow x^2 = 189$$ 6. Simplify $x^2$: $$x = \sqrt{189} = \sqrt{9 \times 21} = 3 \sqrt{21}$$ 7. Notice $6 \sqrt{3} \approx 10.392$ and $3 \sqrt{21} \approx 13.747$, so this conflicts. Because the hypotenuse 15 is not equal to $\sqrt{x^2 + 6^2}$ with $x=6\sqrt{3}$, we must use Pythagorean theorem consistently. 8. Hence correct $x$ (opposite side) is: $$x = 3 \sqrt{21}$$ Final answer in surd form: $x = 3 \sqrt{21}$