Question

if x+1/x=5 find x3-1/x3

Answer 1

Answer:

$\red {x^{3} - \frac{1}{x^{3}} } \green { = 24\sqrt{21} }$

Step-by-step explanation:

$Given \: x + \frac{1}{x} = 5 \: ---(1)$

$\left( x - \frac{1}{x} \right)^{2} = \left( x + \frac{1}{x} \right)^{2} - 4$

$= 5^{2} - 4 \: [ From \: (1) ]$

$= 25 - 4 \\= 21$

$\left( x - \frac{1}{x} \right) = \sqrt{ 21} \: ---(2)$

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

$\boxed { \pink { a^{3} - b^{3} = (a - b)^{3} + 3ab(a-b) }}$

$= ( \sqrt{21} )^{3} + 3\sqrt{21} \: [From \: (2) ]$

$= 21\sqrt{21} + 3\sqrt{21} \\= 24\sqrt{21}$

Therefore.,

$\red {x^{3} - \frac{1}{x^{3}} } \green { = 24\sqrt{21} }$

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