Simplification of surd expression

Expanding the given equation: $(a + b\sqrt{3})^2 = a^2 + 3b^2 + 2ab\sqrt{3} = 52 + 30\sqrt{3}$

Comparing rational and irrational parts: $2ab = 30 \implies ab = 15$ $a^2 + 3b^2 = 52$

Since $a$ and $b$ are natural numbers, the possible pairs for $(a, b)$ are:

• If $a = 5, b = 3$: $a^2 + 3b^2 = 5^2 + 3(3^2) = 25 + 27 = 52$. This satisfies both conditions.

Hence, $a = 5$ and $b = 3$, giving $a + b = 5 + 3 = 8$.