Question

If x = 1 - √2 , find the value of{x-1/x}^3​

Answer 1

Step-by-step explanation:

$\huge \mathcal \red{A}\huge \mathcal \green{n} \huge \mathcal \orange{s}\huge \mathcal \blue{w}\huge \mathcal{e}\huge \mathcal \green{r - }$

Given -

$x = 1 - \sqrt{2}$

To Find -

$(x + \frac{1}{x} ) {}^{3}$

As we know,

$(a + b) {}^{3} = {a}^{3} - {b}^{3} - 3 {a}^{2} b + 3b {}^{2} a$

Now,

According to the question,

$(x + \frac{1}{x} ) {}^{3} \\ \\ {(1 - \sqrt{2})}^{3} - \frac{1}{(1 - \sqrt{2}) {}^{3} } + 3 \times \frac{1}{ {(1 - \sqrt{2})}^{2} } \times (1 - \sqrt{2}) - 3 \times {(1 - \sqrt{2})}^{2} \times \frac{1}{(1 - \sqrt{2})} \\ \\ 7 - 5 \sqrt{2} - \frac{1}{7 - 5 \sqrt{2} } + \frac{3}{(1 - \sqrt{2}) } - 3(1 - \sqrt{2} ) \\ \\ \frac{(7 - 5 \sqrt{2}) {}^{2} - 1}{7 - 5 \sqrt{2} } + 3( \frac{1}{(1 - \sqrt{2} )} - (1 - \sqrt{2} )) \\ \\ \frac{49 + 50 - 70 \sqrt{2} - 1}{7 - 5 \sqrt{2} } + 3( \frac{1 - (1 - \sqrt{2} ) {}^{2} }{1 - \sqrt{2} } ) \\ \\ \frac{98 - 70 \sqrt{2} }{7 - 5 \sqrt{2} } + 3( \frac{1 - (1 + 2 - 2 \sqrt{2}) }{1 - \sqrt{2} } ) \\ \\ \frac{14(7 - 5 \sqrt{2} )}{(7 - 5 \sqrt{2}) } + 3( \frac{ - 2 + 2 \sqrt{2} }{(1 - \sqrt{2}) } ) \\ \\ 14 + 3( \frac{ - 2(1 - \sqrt{2}) }{(1 - \sqrt{2}) } ) \\ \\ 14 + 3 \times - 2 \\ \\ 14 - 6 \\ \\ = 8$

Answer 2

8

Step-by-step explanation:

Given:

x = 1 - √2

To find:

(x - 1/x)³

Solution:

$x = 1 - \sqrt{2} \\ \\ \frac{1}{x} = \frac{1}{1 - \sqrt{2} }$

Now, Rationalizing the denominator ,

$\frac{1}{x} = \frac{(1 + \sqrt{2} )}{(1 - \sqrt{2})(1 + \sqrt{2} )} \\ \\ \frac{1}{x} = \frac{(1 + \sqrt{2}) }{ {1}^{2} - ( \sqrt{2} )^{2} } \\ \\ \frac{1}{x} = \frac{1 + \sqrt{2} }{1 - 2} \\ \\ \frac{1}{x} = \frac{1 + \sqrt{2} }{ - 1} \\ \\ \frac{1}{x} = - 1 - \sqrt{2} \\ \\ - ( \frac{1}{x} ) = 1 + \sqrt{2}$

Now in question,

${(x - \frac{1}{x} )}^{3} \\ \\ \\ \\ (1 - \sqrt{2} + 1 + \sqrt{2} ) {}^{3} \\ \\ {(2)}^{3} \\ \\ 8$

Hence the required value is 8 .