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If $a = 3 + 1 3 − 1 and b = 1 a,$ then which of the following is the value of the expression

$a 2 − b 2 + a b$ ?

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2 + 83

1 + 83

3 + 83

4 + 83

answer is B.

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Detailed Solution

Given that,

a = 3 + 1 3 − 1 and b = 1 a

Use the rationalization formula to rationalize the denominator of

a .

Therefore we have,

a = 3 + 1 3 − 1 × 3 + 1 3 + 1

⇒ (3) 2 + 23 + 12 (3) 2 − 12

⇒ 3 + 23 + 1 3 − 1

⇒ 4 + 23 2

⇒ 2 (2 + 3) 2

⇒ 2 + 3

We have,

b = 1 a

⇒ b = 1 2 + 3

Use the rationalization formula to rationalize the denominator.

⇒ b = 1 2 + 3 × 2 − 3 2 − 3

⇒ 2 − 3 22 − 32 ⇒ 2 − 3 4 − 3 ⇒ 2 − 3

Substitute the values of

a and b

into the expression

a 2 − b 2 + a b .

Therefore we have,

⇒ a 2 − b 2 + a b = (2 + 3) 2 − (2 − 3) 2 + (2 + 3) × 1 (2 + 3)

⇒ (22 + (3) 2 + 2 × 23) − (22 + (3) 2 − 2 × 23) + 1

⇒ 4 + 3 + 43 − 4 − 3 + 43 + 1

⇒ 1 + 83

Therefore, the value of the expression

a 2 − b 2 + a b

is equal to

1 + 83 .

Hence, option (2) is correct.

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