Question

What is the following quotient?

StartFraction StartRoot 6 EndRoot + StartRoot 11 EndRoot Over StartRoot 5 EndRoot + StartRoot 3 EndRoot EndFraction
StartFraction StartRoot 30 EndRoot + 3 StartRoot 2 EndRoot + StartRoot 55 EndRoot + StartRoot 33 EndRoot Over 8 EndFraction
StartFraction StartRoot 30 EndRoot minus 3 StartRoot 2 EndRoot + StartRoot 55 EndRoot minus StartRoot 33 EndRoot Over 2 EndFraction
Seventeen-eighths
Negative five-halves

Answer 1

Which quotient do you mean? If you mean
(√6 + √11) / (√5 + √3), rationalize the denominator:

Multiply top and bottom by (√5 − √3):
(√6 + √11)(√5 − √3) / (5 − 3)
= (√30 − 3√2 + √55 − √33) / 2.

So the exact value is (√30 − 3√2 + √55 − √33)/2 ≈ 1.453110364. That is neither 17/8 (≈2.125) nor −5/2 (−2.5).