Question

If x=4+root 15,find the value of xcube -1upon xcube

Answer 1

$\huge{\bigstar}{ \green{ \mathfrak{ \underline{ \underline{Question}}}}}{\bigstar}$

□☆□☆□☆□☆□☆□☆□☆□☆□☆□☆□☆

$If \ x = 4 + \sqrt15 ,\ find\ the \ value \ of\\ x^3 - \displaystyle\frac{1}{x^3}$

□☆□☆□☆□☆□☆□☆□☆□☆□☆□☆□☆

─────────────────────

$\huge{\bigstar}{ \green{ \mathfrak{ \underline{ \underline{Answer}}}}}{\bigstar}$

$\boxed{\red{\bold{Given:}}}$

$\red{\implies x = 4 +\sqrt15}$

$\red{\implies} \displaystyle\frac{1}{x} = \frac{1}{4+ \sqrt15 }$

$\boxed{\red{\bold{Rationalizing:}}}$

$\purple{\implies x = \displaystyle\frac{1}{4+ \sqrt15 } × \frac{4- \sqrt15 }{4- \sqrt15}}$

$\purple{\implies} \displaystyle \frac {4- \sqrt15 }{4^2 - \sqrt15^2}$

$\purple{\implies} \displaystyle \frac{4- \sqrt15 }{16-15}$

$\purple{\implies}4 - \sqrt15$

$\boxed{\red{\bold{Now :}}}$

$\green{\implies} x^3 - \displaystyle\frac{1}{x^3} = \left( x - \frac{1}{x} \right)^3 + 3 \left( x - \frac{1}{x} \right)$

$\boxed{\red{\bold{Putting \ the \ values:}}}$

$\green{\implies} x^3 - \displaystyle\frac{1}{x^3} = (4 + \sqrt15 - 4 + \sqrt15 )^3 -3(4 + \sqrt15 - 4 + \sqrt15 )$

$\green{\implies} x^3 - \displaystyle\frac{1}{x^3} = ({2\sqrt15})^3 - 3(2\sqrt15 )$

$\green{\implies} x^3 - \displaystyle\frac{1}{x^3} = 1800 \sqrt15 - 6\sqrt15$

$\green{\boxed{\implies{\boxed{x^3 - \displaystyle\frac{1}{x^3} = 1794\sqrt15}}}}$

─────────────────────