1/√5 - 4/√5 - 1 + 1/2 + √5

To simplify the expression 1/√5 - 4/√5-1 + 1/2+√5, we need to rationalize the denominators.

1/√5 can be rationalized by multiplying both the numerator and denominator by √5:

(1/√5) * (√5/√5) = √5/5

Similarly, 4/√5-1 can be rationalized by multiplying both the numerator and denominator by √5+1:

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

Lastly, 1/2+√5 does not require rationalization.

Now we can simplify the expression:

√5/5 - √5 - 1 + √5 + 1/2

The √5 and -√5 terms cancel each other out:

√5/5 - 1 + 1/2

To add these fractions, we need a common denominator, which is 10:

(2√5 + 10 - 5)/10

Simplifying further:

(2√5 + 5)/10

This can be simplified by dividing both the numerator and denominator by 5:

(2√5 + 5)/10 = (2/10)√5 + 5/10 = (1/5)√5 + 1/2

Therefore, the simplified expression is (1/5)√5 + 1/2.