Find the value of the following expression:

Two identities are used here: (1) simplifying a surd of the form √(a2·b) = a√b, and similarly, the cube root of a perfect cube reduces directly to its cube-root value; and (2) rationalizing a denominator of the form 1/(√x − √y) by multiplying numerator and denominator by the conjugate (√x + √y), using the identity (√x − √y)(√x + √y) = x − y.

1. Simplify each root term individually: √20 = √(4×5) = 2√5; √12 = √(4×3) = 2√3; the cube root of 27 = 3; √25 = 5.

2. Rationalize the fraction by multiplying by the conjugate: 4/(√5−√3) = [4(√5+√3)] / [(√5−√3)(√5+√3)] = [4(√5+√3)] / (5−3) = [4(√5+√3)] / 2 = 2√5 + 2√3.

3. Substitute all the simplified terms back into the original expression: 2√5 + 2√3 + 3 − (2√5 + 2√3) − 5.

4. Group like terms: the surd terms cancel out completely, since (2√5 − 2√5) = 0 and (2√3 − 2√3) = 0, leaving only the constant terms 3 − 5.

5. Compute the remaining constants: 3 − 5 = −2.

As a numerical check: √20 is approximately 4.47, √12 is approximately 3.46, the cube root of 27 is exactly 3, √5 − √3 is approximately 0.504 so 4 divided by that is approximately 7.94, and √25 is 5. Combining with the given signs: 4.47 + 3.46 + 3 − 7.94 − 5 is approximately −2.01, which matches the exact algebraic result.

So the value of the expression is −2.