Find the value of ¼ (x3 – 1/x3)

If x – 1/x = 3 + 2√2, find the value of ¼ (x³ – 1/x³)

Answer :

It is given that,

x – 1/x = 3 + 2√2

So,

x³ – 1/x³ = (x – 1/x)³ + 3(x – 1/x)

= (3 + 2√2)³ + 3(3 + 2√2)

By using the formula, (a+b)³ = a³ + b³ + 3ab (a + b)

= (3)³ + (2√2)³ + 3 (3) (2√2) (3 + 2√2) + 3(3 + 2√2)

= 27 + 16√2 + 54√2 + 72 + 9 + 6√2

= 108 + 76√2

Hence,

¼ (x³ – 1/x³) = ¼ (108 + 76√2)

= 27 + 19√2

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