Question

please help me to everyone and solve it question class 9th ​

Answer 1

GIVEN :-

• x = 3 + 2√2.

TO FIND :-

• The value of (x - 1/x)³

SOLUTION :-

$\\ : \implies \displaystyle \sf \: \bigg(x - \frac{1}{x} \bigg)^{3} = \bigg(3 + 2 \sqrt{2} - \frac{1}{3 + 2 \sqrt{2} } \bigg) ^{3}$

$: \implies \displaystyle \sf \: \bigg(x - \frac{1}{x} \bigg)^{3} = \bigg(3 + 2 \sqrt{2} - \frac{1(3 - 2 \sqrt{2} )}{(3 + 2 \sqrt{2} )(3 - 2 \sqrt{2} )} \bigg) ^{3}$

$: \implies \displaystyle \sf \: \bigg(x - \frac{1}{x} \bigg)^{3} = \bigg(3 + 2 \sqrt{2} - \frac{3 - 2 \sqrt{2} }{(3) ^{2} - (2 \sqrt{2} )^{2} } \bigg) ^{3}$

$: \implies \displaystyle \sf \: \bigg(x - \frac{1}{x} \bigg)^{3} = \bigg(3 + 2 \sqrt{2} - \frac{3 - 2 \sqrt{2} }{9 - 4 \times 2} \bigg) ^{3}$

$: \implies \displaystyle \sf \: \bigg(x - \frac{1}{x} \bigg)^{3} = \bigg(3 + 2 \sqrt{2} - \frac{3 - 2 \sqrt{2} }{1} \bigg) ^{3}$

$: \implies \displaystyle \sf \: \bigg(x - \frac{1}{x} \bigg)^{3} = \bigg(3 + 2 \sqrt{2} - 3 + 2 \sqrt{2 } \bigg) ^{3}$

$: \implies \displaystyle \sf \: \bigg(x - \frac{1}{x} \bigg)^{3} = \bigg(4\sqrt{2}\bigg) ^{3}$

$: \implies \displaystyle \sf \: \bigg(x - \frac{1}{x} \bigg)^{3} = 64\sqrt{8}$

$: \implies \underline{ \boxed{\displaystyle \sf \bold{\: \bigg(x - \frac{1}{x} \bigg)^{3} =128\sqrt{2} }}}$

Answer 2

$\sf \underline{Given } :$

• → x = 3 + 2√2 .

$\sf \underline{To \: Find } :$

• ( x - 1/x )³

$\sf \underline{ Solution \: } :$

$\sf : \implies \: then \: \: \frac{1}{x} = \frac{1}{3 + 2 \sqrt{2} } \\ \\ \\ \sf : \implies \: \: \frac{1}{3 + 2 \sqrt{2} } \: \times \frac{3 + 2 \sqrt{2} }{3 + 2 \sqrt{2} } \\ \\ \\ \sf : \implies \: \: \cancel{ \frac{3 + 2 \sqrt{2} \: }{ {3}^{2} + {2 \sqrt{2} }^{2} } }\\ \\ \\ \sf \implies \: \: \: 3 + 2 \sqrt{2}$

$\underline\boldsymbol{According \: to \: the \: question \: }$

$\therefore \sf \: {\frac{x - 1}{x}}^{3} \:$

Substitute all values :

$\sf \implies \: {(3 + 2 \sqrt{2} - 3 + 2 \sqrt{2} })^{3} \\ \\ \\ \sf \implies \: \: {(4 \sqrt{2} })^{3} \\ \\ \: \\ \sf \implies \: \:128 \sqrt{2}$