If (a +b√3)² = 52+30√3, where a and b are natural numbers, then a+b equals:

The correct option is 8.

We are given the equation:
$${(a+b\sqrt{3})}^2=52+30\sqrt{3}$$
where a and b are natural numbers. We want to find the value of a + b.

First, let's expand the left-hand side of the equation using the algebraic identity ${(x+y)}^2=x^2+2xy+y^2$:
$${(a+b\sqrt{3})}^2=a^2+2\cdot a\cdot \left(b\sqrt{3}\right)+{\left(b\sqrt{3}\right)}^2$$

Simplifying the terms, we get:
$${(a+b\sqrt{3})}^2=a^2+2ab\sqrt{3}+3b^2$$

Grouping the rational and irrational parts together, we get:
$${(a+b\sqrt{3})}^2=(a^2+3b^2)+2ab\sqrt{3}$$

Now, we equate this expression to the given value on the right-hand side of our original equation:
$$(a^2+3b^2)+2ab\sqrt{3}=52+30\sqrt{3}$$

By comparing the rational parts and the coefficients of the irrational parts (since a and b are natural numbers), we obtain a system of two equations:
1) $a^2+3b^2=52$
2) $2ab=30$

From equation (2), we can simplify to find the product of a and b:
$$ab=15$$

Since a and b must be natural numbers, we list the possible factor pairs of 15:
• Case 1: a = 1, b = 15
• Case 2: a = 3, b = 5
• Case 3: a = 5, b = 3
• Case 4: a = 15, b = 1

Now, we test these pairs in equation (1), $a^2+3b^2=52$, to find which one satisfies it:

• For Case 1 (a = 1, b = 15):
$1^2+3\cdot {15}^2=1+3\cdot 225=1+675=676\ne 52$ (Incorrect)

• For Case 2 (a = 3, b = 5):
$3^2+3\cdot 5^2=9+3\cdot 25=9+75=84\ne 52$ (Incorrect)

• For Case 3 (a = 5, b = 3):
$5^2+3\cdot 3^2=25+3\cdot 9=25+27=52$ (Correct!)

Thus, the values of the natural numbers are a = 5 and b = 3.

Finally, we calculate a + b:
$$a+b=5+3=8$$