Question

If x=3-√8 then x3+1/x3=?
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Answer 1

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☸ Question:

☛If x=3-√8 then x3+1/x3=.........?

☸ Answer:

☛Given data is as follows:

$\star{ \large{ \sf{ \boxed{ \purple{ \: \: x = 3 - \sqrt{8}}}.....equation(1)}}} \\ \\ ⇢ \large{ \sf{ \frac{1}{x} = \frac{1}{3 - \sqrt{8} } }} \\ \sf{ \underline{ \underline{ \dag{ \green{ \: \: rationalizing \: the \: denominator}}}}} \\ \\ ⇢ \large{ \sf{ \frac{1}{x} = \frac{3 + \sqrt{8} }{(3 - \sqrt{8})(3 + \sqrt{8} )} }} \\ \\ ⇢ \large{ \sf{ \frac{1}{x} = \frac{3 + \sqrt{8} }{ \: \: {3}^{2} - {( \sqrt{8}) }^{2} } }} \\ \\ ⇢ \large{ \sf{ \frac{1}{x} = \frac{3 + \sqrt{8} }{9 - 8} }} \\ \\ ⇢ \large{ \sf{ \frac{1}{x} = 3 + \sqrt{8} ......equation(2)}}$

☛Adding equation (1) and equation (2)$⇢\large{ \sf{x + \frac{1}{x} = 3 \cancel{- \sqrt{8} }+ 3 + \cancel{ \sqrt{8} }}} \\ \\ ⇢ \large{ \sf{x + \frac{1}{x} = 6.........equation(3)}}$

☛We have to find, x3+1/x3

$\large{ \sf{ \star{ we \: know.....}}} \\ \\ \large{ \boxed{ \purple{ \sf{ {a}^{3} + {b}^{3} = (a + b) {}^{3} - 3ab(a + b)}}}} \\ \\ \large{ \sf{ \star{by \: using \: this \: identity....}}} \\ \\ ⇢ \large{ \sf{ {x}^{3} + \frac{1}{ {x}^{3} } = (x + \frac{1}{x} ) {}^{3} - 3 \times \cancel{x }\times \frac{1}{ \cancel{x}} (x + \frac{1}{x} )}} \\ \\ ⇢ \large{ \sf{ {x}^{3} + \frac{1}{ {x}^{3} } = (x + \frac{1}{x} ) {}^{3} - 3(x + \frac{1}{x} )}} \\ \\ \large{ \sf{ \star{by \: putting \: value \: obtained \: in \: equation(3)}}} \\ \\ ⇢ \large{ \sf{ {x}^{3} + \frac{1}{ {x}^{3} } = (6) {}^{3} - 3(6)}} \\ \\ ⇢ \large{ \sf{ {x}^{3} + \frac{1}{ {x}^{3} } = 216 - 18}} \\ \\ ⇢ \large{ \boxed{ \pink{ \sf{ {x}^{3} + \frac{1}{ {x}^{3} } = 198}}}}$

☸ So Final Answer:

$\huge{ \boxed{ \bold{ \red{ = 198}}}}$

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Answer 2

${ \sf{ \underline{✒Given:-}}}$

$x = 3 - \sqrt{8}$

${ \sf{ \underline{✒To \: Find:-}}}$

$Value \: of \: {x}^{3} + \frac{1}{ {x}^{3} }$

${ \sf { \underline{✒Solution:-}}}$

✏$\frac{1}{x } = \frac{1}{3 - \sqrt{8} } \times \frac{3 + \sqrt{8} }{3 + \sqrt{8} }$

✏$\frac{1}{x} = \frac{3 + \sqrt{8} }{ {3}^{2} - { \sqrt{8} }^{2} }$

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

✏$\frac{1}{x} = 3 + \sqrt{8}$

Now,

✏$\frac{x + 1}{x} = 3 - \sqrt{8} + 3 + \sqrt{8}$

Hence,

✏$\frac{x + 1}{y} = 6$

Cubing both sides we get :-

✏$( \frac{x + 1}{x} )^{3} = {6}^{3}$

${x}^{3} + \frac{1}{ {x}^{3} } + 3(6) = 216$

${ \bf{ \boxed{ \pink{ {x}^{3} + \frac{1}{ {x}^{3} } = 198}}}}$.

❃SO THE REQUIRED ANSWER IS 198 .❃

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