Given Equation: $\frac{30}{5\sqrt{3} - 3\sqrt{5}} = a\sqrt{3} + b\sqrt{5}$

To find the values of a and b, we need to rationalize the denominator of the given equation.

Rationalizing the Denominator:
We can multiply the numerator and denominator by the conjugate of the denominator to eliminate the square roots in the denominator.

The conjugate of $5\sqrt{3} - 3\sqrt{5}$ is $5\sqrt{3} + 3\sqrt{5}$.

Multiplying the numerator and denominator by $5\sqrt{3} + 3\sqrt{5}$, we get:

$\frac{30}{5\sqrt{3} - 3\sqrt{5}} \cdot \frac{5\sqrt{3} + 3\sqrt{5}}{5\sqrt{3} + 3\sqrt{5}}$

Simplifying the numerator and denominator:

$= \frac{30(5\sqrt{3} + 3\sqrt{5})}{(5\sqrt{3})^2 - (3\sqrt{5})^2}$

$= \frac{30(5\sqrt{3} + 3\sqrt{5})}{75 - 45}$

$= \frac{30(5\sqrt{3} + 3\sqrt{5})}{30}$

$= 5\sqrt{3} + 3\sqrt{5}$

Comparing with the Given Equation:
From the given equation, we have:

$a\sqrt{3} + b\sqrt{5} = 5\sqrt{3} + 3\sqrt{5}$

Comparing the coefficients of $\sqrt{3}$ and $\sqrt{5}$ on both sides, we get:

$a = 5$ and $b = 3$

Final Answer:
The values of a and b are 5 and 3, respectively.