Question

Please solve the question

Answer 1

Question:

If $\displaystyle{\sf\:x\:=\:7\:+\:4\:\sqrt{3}}$, find the value of $\displaystyle{\sf\:x^3\:+\:\dfrac{1}{x^3}}$

Answer:

$\displaystyle{\boxed{\red{\sf\:x^3\:+\:\dfrac{1}{x^3}\:=\:2702}}}$

Step-by-step-explanation:

We have given that,

$\displaystyle{\sf\:x\:=\:7\:+\:4\:\sqrt{3}}$

We have to find the value of

$\displaystyle{\sf\:x^3\:+\:\dfrac{1}{x^3}}$

Now,

$\displaystyle{\sf\:x\:=\:7\:+\:4\:\sqrt{3}}$

$\displaystyle{\implies\sf\:\dfrac{1}{x}\:=\:\dfrac{1}{7\:+\:4\:\sqrt{3}}}$

$\displaystyle{\implies\sf\:\dfrac{1}{x}\:=\:\dfrac{1}{7\:+\:4\:\sqrt{3}}\:\times\:\dfrac{7\:-\:4\:\sqrt{3}}{7\:-\:4\:\sqrt{3}}\:\:\qquad\:\dots\:[\:Rationalising\:the\:denominator\:]}$

$\displaystyle{\implies\sf\:\dfrac{1}{x}\:=\:\dfrac{7\:-\:4\:\sqrt{3}}{(\:7\:)^2\:-\:(\:4\:\sqrt{3}\:)^2}\:\quad\:\dots\:[\:(\:a\:+\:b\:)\:(\:a\:-\:b\:)\:=\:a^2\:-\:b^2\:]}$

$\displaystyle{\implies\sf\:\dfrac{1}{x}\:=\:\dfrac{7\:-\:4\:\sqrt{3}}{49\:-\:16\:\times\:3}}$

$\displaystyle{\implies\sf\:\dfrac{1}{x}\:=\:\dfrac{7\:-\:4\:\sqrt{3}}{49\:-\:48}}$

$\displaystyle{\implies\sf\:\dfrac{1}{x}\:=\:\dfrac{7\:-\:4\:\sqrt{3}}{1}}$

$\displaystyle{\implies\boxed{\pink{\sf\:\dfrac{1}{x}\:=\:7\:-\:4\:\sqrt{3}}}}$

Now,

$\displaystyle{\sf\:x\:+\:\dfrac{1}{x}\:=\:7\:+\:4\:\sqrt{3}\:+\:7\:-\:4\:\sqrt{3}}$

$\displaystyle{\implies\sf\:x\:+\:\dfrac{1}{x}\:=\:7\:+\:7\:+\:\cancel{4\:\sqrt{3}}\:-\:\cancel{4\:\sqrt{3}}}$

$\displaystyle{\implies\sf\:x\:+\:\dfrac{1}{x}\:=\:7\:+\:7}$

$\displaystyle{\implies\boxed{\purple{\sf\:x\:+\:\dfrac{1}{x}\:=\:14}}}$

We have to find the value of

$\displaystyle{\sf\:x^3\:+\:\dfrac{1}{x^3}}$

$\displaystyle{\sf\:x\:+\:\dfrac{1}{x}\:=\:14}$

$\displaystyle{\implies\sf\:\left(\:x\:+\:\dfrac{1}{x}\:\right)^3\:=\:(\:14\:)^3\:\quad\:\dots\:[\:Cubing\:both\:sides\:]}$

$\displaystyle{\implies\sf\:x^3\:+\:3\:\times\:x^{\cancel{2}}\:\times\:\dfrac{1}{\cancel{x}}\:+\:3\:\times\:x\:\times\:\left(\:\dfrac{1}{x}\:\right)^2\:+\:\left(\:\dfrac{1}{x}\:\right)^3\:=\:2744}$

$\displaystyle{\implies\sf\:x^3\:+\:3x\:+\:3\:\times\:\cancel{x}\:\times\:\dfrac{1}{x^{\cancel{2}}}\:+\:\dfrac{1}{x^3}\:=\:2744}$

$\displaystyle{\implies\sf\:x^3\:+\:3x\:+\:\dfrac{3}{x}\:+\:\dfrac{1}{x^3}\:=\:2744}$

$\displaystyle{\implies\sf\:x^3\:+\:\dfrac{1}{x^3}\:+\:3\:\left(\:x\:+\:\dfrac{1}{x}\:\right)\:=\:2744}$

$\displaystyle{\implies\sf\:x^3\:+\:\dfrac{1}{x^3}\:+\:3\:\times\:14\:=\:2744}$

$\displaystyle{\implies\sf\:x^3\:+\:\dfrac{1}{x^3}\:+\:42\:=\:2744}$

$\displaystyle{\implies\sf\:x^3\:+\:\dfrac{1}{x^3}\:=\:2744\:-\:42}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:x^3\:+\:\dfrac{1}{x^3}\:=\:2702}}}}$