If x = (√7 − √5) / (√5 + √7) and y is the reciprocal of x, then what is the…

Step 1 — Rationalise x.

x = (√7 − √5) / (√7 + √5). Multiply numerator and denominator by (√7 − √5):

x = (√7 − √5)² / [(√7 + √5)(√7 − √5)] = (7 + 5 − 2√35) / (7 − 5) = (12 − 2√35) / 2 = 6 − √35.

Step 2 — Find y.

y is the reciprocal of x, so y = 1/x = (√7 + √5) / (√7 − √5) = (12 + 2√35) / 2 = 6 + √35.

(Check: xy = (6 − √35)(6 + √35) = 36 − 35 = 1, confirming y = 1/x.)

Step 3 — Use x + y and xy.

x + y = (6 − √35) + (6 + √35) = 12 and xy = 1.

Step 4 — Compute x³ + y³.

x³ + y³ = (x + y)³ − 3xy(x + y) = 12³ − 3(1)(12) = 1728 − 36 = 1692.

Step 5 — Take the square root.

√(x³ + y³) = √1692 = √(36 × 47) = 6√47.

Hence the value is 6√47.