Let \(\sqrt{3+2\sqrt{10}i}\) = a + bi, where a, b ∈ R.

**Squaring on both sides, we get**

3 + 2√10 i = a^{2} + b^{2} i^{2}+ 2abi

3 + 2√10 i = (a^{2} – b^{2} ) + 2abi …**…[∵ i**^{2} = -1]

**Equating real and imaginary parts, we get**

a^{2} – b^{2} = 3 and 2ab = 2√10