Question

If
$x + \frac{1}{x} = 8$
then find the value of
$x^{3} + \frac{1}{ {x}^{3} }$
​

Answer 1

Answer :-

• The value of $\sf x^{3} +\sf\cfrac{1}{{x}^{3}}$ = 488

To Find :-

• The value of $\sf x^{3} +\cfrac{1}{{x}^{3}}$

Given :-

• $\sf x + \cfrac{1}{x} = 8$

Step By Step Explanation :-

We know that $\sf{x + \cfrac{1}{x} = 8}$

We need to calculate the value of $\sf x^{3} +\cfrac{1}{{x}^{3}}$

So let's do it !!

$\bf \: Using \: Identity \downarrow \\ \\\dag\boxed{ \bf{ \red {{(x + y)}^{3}= {x}^{3} + {y}^{3} + 3xy(x + y)}}}$

By substituting the values ⤵

$\sf \left(\cfrac{1}{x}+x\right)^{3}= \cfrac{1}{ {x}^{3} } + {x}^{3} + 3 \times \cfrac{1}{ \not x} \times \not x \: \left( x + \cfrac{1}{x}\right) \\ \\ \bf \: By \: substituting \: the \: values \downarrow \\ \\\implies \sf {(8)}^{3} = \cfrac{1}{ {x}^{3} } + {x}^{3} + 3(8) \\ \\\implies \sf 512 = \cfrac{1}{ {x}^{3} } + {x}^{3} + 24 \\ \\ \implies \sf512 - 24 = \cfrac{1}{ {x}^{3} } + {x}^{3} \\ \\ \implies\bf 488 = \cfrac{1}{ {x}^{3} } + {x}^{3}$

Hence the value of $\sf x^{3} +\cfrac{1}{{x}^{3}}$ = 488

__________________________

Answer 2

Answer:

Given

$x + \frac{1}{x} = 8$

To find

${x}^{3} + \frac{1}{ {x}^{3} }$

solve

$(x + y) {}^{3} = x {}^{3} + y {}^{3} + 3xy \: (x + y)$

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

By Substitution the value

$({8}^{3}) = \frac{1}{ {x}^{3} } + {x}^{3} + {x}^{3} + 3(8)$

$512 = \frac{1}{x {}^{3} }+ {x}^{3} + 24$

$512 - 24 = \frac{1}{ {x}^{3} } + {x}^{3}$

$488 = \frac{1}{ {x}^{3} } + {x}^{3}$

Hence Answer is

$488 = \frac{1}{ {x}^{3} } + {x}^{3}$