Question

if x = 3+2 under root 2 so find the value of (x-1/x) ^3

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Answer 1

Answer :-

The value of (x - 1/x)³ is 128√2.

Solution :-

x = 3 + 2√2

Reciprocal on both sides

1/x = 1/(3 + 2√2)

Rationalise the denominator

$\\ \\ \tt = \dfrac{1}{3 + 2 \sqrt{2} } \times \dfrac{3 - 2 \sqrt{2} }{3 - 2 \sqrt{2} } \\ \\ \\ \tt = \dfrac{3 - 2 \sqrt{2} }{ {3}^{2} - {(2 \sqrt{2})}^{2} } \\ \\ \\ \tt = \dfrac{3 - 2 \sqrt{2} }{9 - 4(2)} \\ \\ \\ \tt = \frac{3 - 2 \sqrt{2} }{9 - 8} \\ \\ \\ \tt = \frac{3 - 22}{1} \\ \\ \\ \tt = 3 - 2 \sqrt{2} \\ \\ \\ \bf \therefore \dfrac{1}{x} = 3 - 2 \sqrt{2}$

Now find the value of x - 1/x

x - 1/x = 3 + 2√2 - (3 - 2√2)

⇒ x - 1/x = 3 + 2√2 - 3 + 2√2

⇒ x - 1/x = 4√2

Cubing on both sides

⇒ (x - 1/x)³ = (4√2)³

⇒ (x - 1/x)³ = 64(2√2)

⇒ (x - 1/x)³ = 128√2

Therefore the value of (x - 1/x)³ is 128√2.