1. The problem is to find the hypotenuse $a$ of a right triangle where the legs are $\sqrt{10}$ and $\sqrt{6}$.\n\n2. According to the Pythagorean theorem, for a right triangle with legs $b$ and $c$, and hypotenuse $a$, the relationship is: $$a^2 = b^2 + c^2$$\n\n3. Substitute the given leg lengths: $$a^2 = (\sqrt{10})^2 + (\sqrt{6})^2$$\n\n4. Simplify the squares of the square roots: $$a^2 = 10 + 6$$\n\n5. Add the values: $$a^2 = 16$$\n\n6. To find $a$, take the square root of both sides: $$a = \sqrt{16}$$\n\n7. Simplify the square root: $$a = 4$$\n\nTherefore, the value of $a$, the hypotenuse, is 4.