Question

x=3+2√2, find the value of x^3 - 1/x^3​

Answer 1

$\huge\bf{\underline{Solution:-}}$

$\sf\bullet x=3+2\sqrt{2}\\ \\ \sf\:\:{\purple{we\:have\:to\: find\:the\: value\:of}}\\ \\ \sf\implies x{}^{2}-\dfrac{1}{x{}^{3}}$

$\sf So\:1st\:find\:the\:value\:of\:\dfrac{1}{x}\\ \\ \sf\leadsto \dfrac{1}{x}=\dfrac{1}{3+2\sqrt{2}}\times \dfrac{3-2\sqrt{2}}{3-2\sqrt{2}}\\ \\ \sf\implies \dfrac{1}{x}=\dfrac{3-2\sqrt{2}}{(3){}^{2}-(2\sqrt{2}){}^{2}}\\ \\ \sf\implies \dfrac{1}{x}=\dfrac{3-2\sqrt{2}}{9-8}\\ \\ \sf\implies {\blue{\dfrac{1}{x}=3-2\sqrt{2}}}$

$\sf\:\:\:\:\:\:\:\:\: identity\:\:used:-\\ \\ \boxed{\sf{(a-b){}^{2}=a{}^{3}-b{}^{3}-3ab(a-b)}}$

$\sf \bigg(x-\dfrac{1}{x}\bigg){}^{3}=x{}^{3}-\dfrac{1}{x{}^{3}}-3\times \cancel{x}\times \dfrac{1}{\cancel{x}}\bigg(x-\dfrac{1}{x}\bigg)\\ \\ \sf\implies x{}^{3}-\dfrac{1}{x{}^{3}}-3\bigg(x-\dfrac{1}{x}\bigg)$

$\sf\:\:Now\:the\:value\:of\: \bigg(x-\dfrac{1}{x}\bigg)\\ \\ \sf\implies 3+2\sqrt{2}-(3-\sqrt{2})\\ \\ \sf\implies \cancel{3}+2\sqrt{2}\cancel{-3}-2\sqrt{2}\\ \\ \sf\implies {\purple{ \bigg(x-\dfrac{1}{x}\bigg)=(-4\sqrt{2})}}$

• Now the Value of

$\sf\bigg( x-\dfrac{1}{x}\bigg){}^{3}=x{}^{3}-\dfrac{1}{x{}^{3}}-3\bigg(x-\dfrac{1}{x}\bigg)\\ \\ \sf\bullet {\pink{Put\:the\: value\:of \:\: x-\dfrac{1}{x}= (-4\sqrt{2})}}\\ \\ \sf\implies (-4\sqrt{2}){}^{3}=x{}^{3}-\dfrac{1}{x{}^{3}}-3(-4\sqrt{2})\\ \\ \sf\implies -128\sqrt{2}=x{}^{3}-\dfrac{1}{x^{3}}+12\sqrt{2}\\ \\ \sf\implies -128\sqrt{2}-12\sqrt{2}=x{}^{3}-\dfrac{1}{x{}^{3}}\\ \\ \sf\implies (-140\sqrt{2})=x{}^{3}-\dfrac{1}{x{}^{3}}$

$\boxed{\mathfrak{\purple{x{}^{3}-\dfrac{1}{x{}^{3}}= (-140\sqrt{2})}}}$

Answer 2

$\red{ \boxed {given}}$

Value of x=3+2√2

$\red {find \: out \:}$

$\: the \: value \: of \: x {}^{3} - \frac{1}{x {}^{3} }$

$\red{solution}$

$\frac{1}{x} = \frac{1}{3 + 2 \sqrt{2} }$

$\frac{1}{x} = \frac{1}{3 + 2 \sqrt{2} } \times \frac{3 - 2 \sqrt{2} }{3 - 2 \sqrt{2} }$

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

$\frac{1}{x} = \frac{3 - 2 \sqrt{2} }{9 - 8}$

$\frac{1}{x} = 3 - 2 \sqrt{2}$

Now,

x-1/x

x-1/x= 3-2√2-(3+2√2)= -4√2

we know that

a³-b³ = (a-b)³+3ab(a-b)

⭐Refers to the attachments