Question

if x-1/x = 3 +2 √2 then find x³ -. 1/x³​

Answer 1

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Answer 2

Given :-

$\rm :\longmapsto\:x - \dfrac{1}{x} = 3 + 2 \sqrt{2}$

To Find :-

$\rm :\longmapsto\: {x}^{3} - \dfrac{1}{ {x}^{3} }$

Solution :-

Given that,

$\rm :\longmapsto\:x - \dfrac{1}{x} = 3 + 2 \sqrt{2}$

We know that,

$\pink{\rm :\longmapsto\: \bf \: {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y)}$

$\pink{\bf\implies \: {x}^{3} - {y}^{3} = {(x - y)}^{3} + 3xy(x - y)}$

Now,

Consider,

$\red{\bf :\longmapsto\: {x}^{3} - \dfrac{1}{ {x}^{3} }}$

$\: \: \rm \: = \: \: {\bigg(x - \dfrac{1}{x} \bigg) }^{3} + 3 \times x \times \dfrac{1}{x} \times \bigg(x - \dfrac{1}{x} \bigg)$

$\: \: \rm \: = \: \: {(3 + 2 \sqrt{2} )}^{3} + 3(3 + 2 \sqrt{2})$

$\: \: \rm \: = \: \:(3 + 2 \sqrt{2})\bigg( {(3 + 2 \sqrt{2} )}^{2} + 3\bigg)$

$\: \: \rm \: = \: \:(3 + 2 \sqrt{2})(9 + 8 + 12 \sqrt{2} + 3)$

$\: \: \rm \: = \: \:(3 + 2 \sqrt{2})(20+ 12 \sqrt{2})$

$\: \: \rm \: = \: \:60 + 36 \sqrt{2} + 40 \sqrt{2} + 48$

$\: \: \rm \: = \: \:108 + 76 \sqrt{2}$

Hence,

$\: \: \: \: \: \: \: \red{\bf :\implies\: {x}^{3} - \dfrac{1}{ {x}^{3} } = 108 + 76 \sqrt{2} }$

Additional Information :-

More Identities to know:

• (a + b)² = a² + 2ab + b²

• (a - b)² = a² - 2ab + b²

• a² - b² = (a + b)(a - b)

• (a + b)² = (a - b)² + 4ab

• (a - b)² = (a + b)² - 4ab

• (a + b)² + (a - b)² = 2(a² + b²)

• (a + b)³ = a³ + b³ + 3ab(a + b)

• (a - b)³ = a³ - b³ - 3ab(a - b)