To simplify the expression E = √(3+√5)² - √5, we can start by expanding the square of (3+√5):

(3+√5)² = (3+√5)(3+√5) = 3(3) + 3(√5) + √5(3) + √5(√5)
= 9 + 3√5 + 3√5 + 5
= 14 + 6√5

Now, the expression becomes:

E = √(14 + 6√5) - √5

To simplify further, we can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator:

E = (√(14 + 6√5) - √5) * (√(14 + 6√5) + √5) / (√(14 + 6√5) + √5)

Using the difference of squares formula, we have:

E = (14 + 6√5 - 5) / (√(14 + 6√5) + √5)
= (9 + 6√5) / (√(14 + 6√5) + √5)

Now, we can simplify the expression by multiplying the numerator and denominator by the conjugate of the denominator again:

E = (9 + 6√5) * (√(14 + 6√5) - √5) / (√(14 + 6√5) - √5) * (√(14 + 6√5) + √5) / (√(14 + 6√5) + √5)

Using the difference of squares formula again, we have:

E = (9 + 6√5)(√(14 + 6√5) - √5) / (14 + 6√5 - 5)
= (9 + 6√5)(√(14 + 6√5) - √5) / (9 + 6√5)

Expanding the numerator:

E = (9√(14 + 6√5) + 6√5√(14 + 6√5) - 9√5 - 6√5) / (9 + 6√5)
= (9√(14 + 6√5) - 9√5) / (9 + 6√5)

Factoring out 3 from the numerator:

E = 3(√(14 + 6√5) - √5) / (9 + 6√5)

Therefore, the simplified expression is E = 3(√(14 + 6√5) - √5) / (9 + 6√5).