Question

if a = 1-√2 find (a-1/a)^3​

Answer 1

This is the answer. Thank you.

Answer 2

Answer:

$\large{\underline{\boxed{\sf (a - \dfrac{1}{a})^3 =8}}}$

Step-by-step explanation:

Given that ;

a = 1 - √2

Find the value of 1/a.

$\implies \sf \frac{1}{a} = \frac{1}{1 - \sqrt{2} } \\ \\ \implies \sf \frac{1}{a} = \frac{1}{1 - \sqrt{2} } \times \frac{1 + \sqrt{2} }{1 + \sqrt{2} } \\ \\ \implies \sf \frac{1}{a} = \frac{1 + \sqrt{2} }{(1) {}^{2} - ( \sqrt{2}) {}^{2} } \\ \\ \implies \sf \frac{1}{a} = \frac{1 + \sqrt{2} }{1 - 2} \\ \\ \implies \sf \frac{1}{a} = \frac{1 + \sqrt{2} }{ - 1} \\ \\ \implies \sf \frac{1}{a} = - (1 + \sqrt{2} )$

Now, find the value of a + (1/a);

$\implies \sf a - \frac{1}{a} = 1 - \sqrt{2} + 1 + \sqrt{2} \\ \\ \implies \sf a - \frac{1}{a} = 1 + 1 \\ \\ \implies \sf a - \frac{1}{a} = 2$

On cubing both sides ;

$\implies \sf (a - \frac{1}{a})^{3} \\ \\ \implies \sf (2) {}^{3} \\ \\ \implies \sf 8$

Hence, the answer is 8.

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

• (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