Question

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Answer 1

Question: If x - 1/x = 3 + 2√2, find the value of x³ - 1/x³.

Answer:

$\large{\underline{\boxed{\sf {x}^{3} - \frac{1}{x^{3}} = 108 + 76 \sqrt{2}}}}$

Step-by-step explanation:

$\large{\underline{\underline{\sf Given \: that :}}}$

$\sf x - \frac{1}{x} = 3 + 2 \sqrt{2} \\ \\ \bf On \: cubing \: both \: sides - \\ \\ \implies \sf (x - \frac{1}{x} ) {}^{3} = {(3 + 2 \sqrt{2}) }^{3} \\ \\ \implies \small \sf {x}^{3} - { \frac{1}{x {}^{3} }} - 3 \:. \: x \: . \: \frac{1}{x} (x - \frac{1}{x} ) = (3) {}^{3} + (2 \sqrt{2} ) {}^{3} + 3 *3*2 \sqrt{2} (3 + 2 \sqrt{2} ) \\ \\ \implies \small \sf {x}^{3} - { \frac{1}{x {}^{3} }} - 3(3 + 2 \sqrt{2} ) = 27+ 16 \sqrt{2} + 18 \sqrt{2}(3 + 2 \sqrt{2} ) \\ \\ \implies \small \sf {x}^{3} - { \frac{1}{x {}^{3} }} - 9 - 6 \sqrt{2} = 27 + 16 \sqrt{2} + 54 \sqrt{2} + 72 \\ \\ \implies \small \sf {x}^{3} - { \frac{1}{x {}^{3} }} = 99 + 70 \sqrt{2} + 9 + 6 \sqrt{2} \\ \\ \implies \small \sf {x}^{3} - { \frac{1}{x {}^{3} }} = 108 + 76 \sqrt{2} \\ \\ \underline{ \boxed{ \bf {x}^{3} - { \frac{1}{x {}^{3} }} = 108+ 76 \sqrt{2}}}$

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$\large{\underline{\underline{\mathfrak{\heartsuit \: Algebraic \: Identities : \: \heartsuit}}}}$

• (a - b)³ = a³ - b³ - 3ab (a - b)
• (a + b)³ = a³ + b³ + 3ab (a + b)
• (a + b)² = a² + 2ab + b²
• (a - b)² = a² - 2ab + b²
• (x + a)(x + b) = x² + x(a + b) + ab