The value of √5 divided by ( √(9 + 4√5) + √(9 − 4√5) ) is

A nested radical of the form √(a ± b√c) can be rewritten as √p ± √q whenever p + q = a and 2√(pq) = b√c, using the identity (√p ± √q)2 = p + q ± 2√(pq). This turns the square root of an irrational sum into a sum or difference of simpler surds.

1. Match 9 + 4√5 with p + q + 2√(pq): here p + q = 9 and 2√(pq) = 4√5, so pq = 20. The pair p = 5, q = 4 satisfies both (5 + 4 = 9 and 5 × 4 = 20).

2. So 9 + 4√5 = (√5 + 2)2, and 9 − 4√5 = (√5 − 2)2 — the same pair with a minus sign gives the difference form.

3. Since √5 ≈ 2.236 is greater than 2, both square roots are of positive quantities: √(9 + 4√5) = √5 + 2 and √(9 − 4√5) = √5 − 2.

4. Add the two roots: √(9 + 4√5) + √(9 − 4√5) = (√5 + 2) + (√5 − 2) = 2√5.

5. The expression becomes √5 ÷ 2√5. The common factor √5 cancels, leaving 1/2.

Cross-check numerically: 9 + 4√5 ≈ 17.944, and √17.944 ≈ 4.236, which matches √5 + 2 = 4.236; 9 − 4√5 ≈ 0.056, and √0.056 ≈ 0.236, which matches √5 − 2 = 0.236. Their sum ≈ 4.472 = 2√5, and √5 ÷ 2√5 ≈ 2.236 ÷ 4.472 ≈ 0.5, confirming the value 1/2.