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

Given:

x = (√7 − √5)/(√5 + √7)

y is reciprocal of x.

So,

y = (√5 + √7)/(√7 − √5)

Now simplify x:

x = (√7 − √5)/(√5 + √7)

Multiply numerator and denominator by (√7 − √5):

x = (√7 − √5)²/(7 − 5)

= (7 + 5 − 2√35)/2

= (12 − 2√35)/2

= 6 − √35

Similarly,

y = 1/x = 6 + √35

Now,

x + y = 12

xy = 1

Using identity:

x³ + y³ = (x + y)³ − 3xy(x + y)

= 12³ − 3(1)(12)

= 1728 − 36

= 1692

Therefore,

√(x³ + y³)

= √1692

= √(36 × 47)

= 6√47