Given the equation $(a + b\sqrt{3})^2 = 52 + 30\sqrt{3}$, where $a$ and $b$ are natural numbers.
Expand the left side:
\[(a + b\sqrt{3})^2 = a^2 + 2ab\sqrt{3} + 3b^2\]
This gives us:
- Rational part: $a^2 + 3b^2$
- Irrational part: $2ab\sqrt{3}$
Equate the rational and irrational parts from both sides of the equation:
1. $a^2 + 3b^2 = 52$
2. $2ab = 30$
From equation 2, $2ab = 30$, we get:
\[ab = 15\]
Substitute $b = \frac{15}{a}$ into equation 1:
\[a^2 + 3\left(\frac{15}{a}\right)^2 = 52\]
Simplify:
\[a^2 + \frac{675}{a^2} = 52\]
Multiply by $a^2$ to remove the denominator:
\[a^4 + 675 = 52a^2\]
Rearrange the equation:
\[a^4 - 52a^2 + 675 = 0\]
Let $x = a^2$. The equation becomes:
\[x^2 - 52x + 675 = 0\]
Solve this quadratic equation using the quadratic formula:
\[x = \frac{52 \pm \sqrt{52^2 - 4 \times 1 \times 675}}{2 \times 1}\]
\[x = \frac{52 \pm \sqrt{2704 - 2700}}{2}\]
\[x = \frac{52 \pm \sqrt{4}}{2}\]
\[x = \frac{52 \pm 2}{2}\]
This gives $x = 27$ or $x = 25$. Since $x = a^2$, we have $a^2 = 25$, which means $a = 5$.
Substitute $a = 5$ into $ab = 15$:
\[5b = 15 \implies b = 3\]
Therefore, $a = 5$ and $b = 3$, so:
\[a + b = 5 + 3 = 8\]
The correct answer is Option (1).