Simplify: √(5+2√6)⁄√(5−2√6)

Correct — B. Each number under a root is a perfect square.

Split 5 as 3 + 2, with 3 × 2 = 6: 5 + 2√6 = (√3 + √2)²
Likewise: 5 − 2√6 = (√3 − √2)²
Take the roots: (√3 + √2)⁄(√3 − √2)

Rationalise, multiplying top and bottom by √3 + √2: (√3 + √2)²⁄(3 − 2)
Expand: (√3 + √2)² = 5 + 2√6 → option (b)

Why the others are wrong

• (a)
  5−2√6 — 5 − 2√6 is the answer upside down: (5 + 2√6)(5 − 2√6) = 25 − 24 = 1, so the two are reciprocals. It is about 0.10, yet the larger root is on top.
• (c)
  3 + 2√3 — 3 + 2√3 ≈ 6.46, but the expression equals (√3 + √2)² = 5 + 2√6 ≈ 9.90. No step of the simplification produces a 2√3 term.
• (d)
  3 − 2√3 — 3 − 2√3 is negative, since 2√3 ≈ 3.46. A ratio of two positive square roots cannot be negative.

A surd like a + 2√b is a perfect square when you can find x and y with x + y = a and xy = b. Then a + 2√b = (√x + √y)², and √(a + 2√b) = √x + √y.

Here 3 + 2 = 5 and 3 × 2 = 6, so 5 + 2√6 = (√3 + √2)² and 5 − 2√6 = (√3 − √2)². Each square root then comes out cleanly.

Key facts

• (√x + √y)² = x + y + 2√(xy).
• 5 + 2√6 = (√3 + √2)² and 5 − 2√6 = (√3 − √2)².
• (5 + 2√6)(5 − 2√6) = 1, so each is the other's reciprocal.

Common traps

• Inverting the fraction and choosing 5 − 2√6.
• Splitting √(5 + 2√6) into √5 + √(2√6), though a square root does not split across a sum.

No directly related past PYQ was found.