To solve for \( a \) in the equation
\[ A = \sqrt{1 + \sqrt{\frac{a}{b}} \]
we will first square both sides to eliminate the square root:
\[ A^2 = 1 + \sqrt{\frac{a}{b}}. \]
Next, we can isolate the square root term:
\[ \sqrt{\frac{a}{b}} = A^2 - 1. \]
Now, we will square both sides again to eliminate the square root:
\[ \frac{a}{b} = (A^2 - 1)^2. \]
Now we will multiply both sides by \( b \) to solve for \( a \):
\[ a = b(A^2 - 1)^2. \]
Thus, the solution for \( a \) is:
\[ a = b(A^2 - 1)^2. \]