Given:
a = (3 sqrt(5))/(3 - sqrt(5)) X
b = (3 - sqrt(5))/(3 sqrt(5))

To find:
a^2 - b^2

Solution:

Step 1: Find the value of a
To find the value of a, we substitute the given expression for a:

a = (3 sqrt(5))/(3 - sqrt(5)) X

Step 2: Simplify the expression for a
To simplify the expression for a, we rationalize the denominator:

a = [(3 sqrt(5))/(3 - sqrt(5))] X [(3 + sqrt(5))/(3 + sqrt(5))]

Using the difference of squares, we get:

a = [(3 sqrt(5))(3 + sqrt(5))]/[(3 - sqrt(5))(3 + sqrt(5))]

Expanding the numerator and denominator, we have:

a = [9 + 3 sqrt(5) + 3 sqrt(5) + 5]/[9 - 3 sqrt(5) + 3 sqrt(5) - 5]

Simplifying further, we get:

a = [14 + 6 sqrt(5)]/4

Dividing both the numerator and denominator by 2, we obtain:

a = (7 + 3 sqrt(5))/2

Step 3: Find the value of b
To find the value of b, we substitute the given expression for b:

b = (3 - sqrt(5))/(3 sqrt(5))

Step 4: Simplify the expression for b
To simplify the expression for b, we rationalize the denominator:

b = [(3 - sqrt(5))/(3 sqrt(5))] X [(3 + sqrt(5))/(3 + sqrt(5))]

Using the difference of squares, we get:

b = [(3 - sqrt(5))(3 + sqrt(5))]/[(3 sqrt(5))(3 + sqrt(5))]

Expanding the numerator and denominator, we have:

b = [9 - 3 sqrt(5) + 3 sqrt(5) - 5]/[9 sqrt(5) + 3 sqrt(5) - 3 sqrt(5) - sqrt(5)]

Simplifying further, we get:

b = [4]/[8 sqrt(5)]

Dividing both the numerator and denominator by 4, we obtain:

b = 1/(2 sqrt(5))

Step 5: Calculate a^2 - b^2
Now that we have the values of a and b, we can calculate a^2 - b^2:

a^2 - b^2 = [(7 + 3 sqrt(5))/2]^2 - [1/(2 sqrt(5))]^2

Expanding the squares, we have:

a^2 - b^2 = [(49 + 42 sqrt(5) + 45)/4] - [1/(4 sqrt(5))]

Simplifying further, we get:

a^2 - b^2 = (49 + 42 sqrt(5) + 45 - 1)/(4)

Combining like terms, we obtain:

a^2 -