\(If a = 8\sqrt{6} - 8\sqrt{5}, b = 8\sqrt{6} + 8\sqrt{5}, c = 6\sqrt{6} + 6\sqrt{5}, d = 4\sqrt{6} + 4\sqrt{5} and e = \sqrt{6} + \sqrt{5}, \)  then which of the following is a rational number?

Given:

a = 8√6 - 8√5

b = 8√6 + 8√5

c = 6√6 + 6√5

d = 4√6 + 4√5

e = √6 + √5

Formula used:

A number is rational if it does not contain any square root (√) term.

Calculations:

Check each pair:

1. a × b:

a × b = (8√6 - 8√5)(8√6 + 8√5)

⇒ a × b = (8√6)² - (8√5)²

⇒ a × b = 64 × 6 - 64 × 5

⇒ a × b = 384 - 320

⇒ a × b = 64 (Rational)

2. c × d:

c × d = (6√6 + 6√5)(4√6 + 4√5)

⇒ c × d = (6√6)(4√6) + (6√6)(4√5) + (6√5)(4√6) + (6√5)(4√5)

⇒ c × d = 24 × 6 + 24√30 + 24√30 + 24 × 5

⇒ c × d = 144 + 120 + 48√30

⇒ c × d contains √30 (Irrational)

3. d × e:

d × e = (4√6 + 4√5)(√6 + √5)

⇒ d × e = (4√6)(√6) + (4√6)(√5) + (4√5)(√6) + (4√5)(√5)

⇒ d × e = 4 × 6 + 4√30 + 4√30 + 4 × 5

⇒ d × e = 24 + 20 + 8√30

⇒ d × e contains √30 (Irrational)

Conclusion:

Among the options, only a × b is rational.

∴ The correct answer is option (3).