Given:
a = 9 - 4√5

To find:
Value of √a - 1/√a

Solution:

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

a = 9 - 4√5

Step 2: Simplify the expression
To simplify the expression, we can rewrite it as follows:

√a - 1/√a = √(9 - 4√5) - 1/√(9 - 4√5)

Step 3: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator:

√a - 1/√a = (√(9 - 4√5) - 1/√(9 - 4√5)) * (√(9 - 4√5) + 1/√(9 - 4√5))

Step 4: Simplify the expression further
Using the difference of squares formula, we can simplify the expression:

√a - 1/√a = (9 - 4√5 + 1) / (√(9 - 4√5) * (√(9 - 4√5) + 1/√(9 - 4√5)))

Simplifying the numerator:

√a - 1/√a = (10 - 4√5) / (√(9 - 4√5) * (√(9 - 4√5) + 1/√(9 - 4√5)))

Step 5: Rationalize the denominator again
To rationalize the denominator, we need to multiply the numerator and denominator by (√(9 - 4√5) - 1/√(9 - 4√5)):

√a - 1/√a = ((10 - 4√5) * (√(9 - 4√5) - 1/√(9 - 4√5))) / ((√(9 - 4√5))^2 - (1/√(9 - 4√5))^2)

Simplifying the denominator:

√a - 1/√a = ((10 - 4√5) * (√(9 - 4√5) - 1/√(9 - 4√5))) / (9 - 2√5 + 1/9 - 2√5)

√a - 1/√a = ((10 - 4√5) * (√(9 - 4√5) - 1/√(9 - 4√5))) / (10 - 4√5 - 4√5 + 1/9)

√a - 1/√a = ((10 - 4√5) * (√(9 - 4