Question

If x2 + 1/x^2 = 66, find the value of x^3 - 1/x^3

Answer 1

$\large\underline{\sf{Solution-}}$

Given expression is

$\rm \: {x}^{2} + \dfrac{1}{ {x}^{2} } = 66$

On Subtracting 2, from both sides, we get

$\rm \: {x}^{2} + \dfrac{1}{ {x}^{2} } - 2= 66 - 2$

$\rm \: {x}^{2} + \dfrac{1}{ {x}^{2} } - 2 \times 1= 64$

$\rm \: {x}^{2} + \dfrac{1}{ {x}^{2} } - 2 \times x \times \frac{1}{x} = 64$

$\rm \: {\bigg(x - \dfrac{1}{x} \bigg) }^{2} = {8}^{2}$

$\rm\implies \:x - \dfrac{1}{x} \: = \: \pm \: 8$

Now, Let Consider

$\rm \: x - \dfrac{1}{x} = 8$

So, on cubing both sides, we get

$\rm \: \bigg(x - \dfrac{1}{x}\bigg)^{3} = {8}^{3}$

$\rm \: {x}^{3} - \dfrac{1}{ {x}^{3} } - 3 \times x \times \dfrac{1}{x}\bigg(x - \dfrac{1}{x} \bigg) = 512$

$\rm \: {x}^{3} - \dfrac{1}{ {x}^{3} } -3 \times 8 = 512$

$\rm \: {x}^{3} - \dfrac{1}{ {x}^{3} } -24 = 512$

$\rm \: {x}^{3} - \dfrac{1}{ {x}^{3} } = 512 + 24$

$\bf\implies \: {x}^{3} - \dfrac{1}{ {x}^{3} } = 536$

Now, Let Consider

$\rm \: x - \dfrac{1}{x} = \: - \: 8$

On cubing both sides, we get

$\rm \: \bigg(x - \dfrac{1}{x}\bigg)^{3} = {( - 8)}^{3}$

$\rm \: {x}^{3} - \dfrac{1}{ {x}^{3} } - 3 \times x \times \dfrac{1}{x}\bigg(x - \dfrac{1}{x} \bigg) = - 512$

$\rm \: {x}^{3} - \dfrac{1}{ {x}^{3} } - 3 \times ( - 8) = - 512$

$\rm \: {x}^{3} - \dfrac{1}{ {x}^{3} } + 24 = - 512$

$\rm \: {x}^{3} - \dfrac{1}{ {x}^{3} } = - 512 - 24$

$\bf\implies \: {x}^{3} - \dfrac{1}{ {x}^{3} } \: = \: - \: 536$

Hence,

$\bf\implies \:\boxed{ \bf{ \: {x}^{3} - \dfrac{1}{ {x}^{3} } \: = \: \pm \: 536 \: \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: identities}}}} \\ \\ \bigstar \: \bf{ {(x + y)}^{2} = {x}^{2} + 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {(x - y)}^{2} = {x}^{2} - 2xy + {y}^{2} }\:\\ \\ \bigstar \: \bf{ {x}^{2} - {y}^{2} = (x + y)(x - y)}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} - {(x - y)}^{2} = 4xy}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{2} + {(x - y)}^{2} = 2( {x}^{2} + {y}^{2})}\:\\ \\ \bigstar \: \bf{ {(x + y)}^{3} = {x}^{3} + {y}^{3} + 3xy(x + y)}\:\\ \\ \bigstar \: \bf{ {(x - y)}^{3} = {x}^{3} - {y}^{3} - 3xy(x - y) }\:\\ \\ \bigstar \: \bf{ {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2} )}\: \end{array} }}\end{gathered}\end{gathered}\end{gathered}$

Answer 2

Answer:

Answer: Our required value is 536.