1. **State the problem:** We have a right triangle with legs of lengths $\sqrt{10}$ and $\sqrt{6}$, and we want to find the length of the hypotenuse. 2. **Formula used:** In a right triangle, the Pythagorean theorem states that the square of the hypotenuse $c$ is equal to the sum of the squares of the legs $a$ and $b$: $$c^2 = a^2 + b^2$$ 3. **Apply the formula:** Here, $a = \sqrt{10}$ and $b = \sqrt{6}$, so $$c^2 = (\sqrt{10})^2 + (\sqrt{6})^2$$ 4. **Simplify the squares:** $$c^2 = 10 + 6$$ 5. **Add the values:** $$c^2 = 16$$ 6. **Find the hypotenuse by taking the square root:** $$c = \sqrt{16}$$ 7. **Simplify the square root:** $$c = 4$$ **Final answer:** The length of the hypotenuse is $4$.