Question

if (x + 1/x)=4

calculate: x^3 - 1/x^3

Answer 1

$\begin{gathered}\large {\boxed{\sf{\mid{\overline {\underline {\star ANSWER ::}}}\mid}}}\end{gathered}$

Given :-

• (x + 1/x) = 4

To Find :-

• The value of x^3 - 1/x^3.

Solution :-

FORMULA :

$\boxed{ \sf{ {(a + b)}^{3} = {a}^{3} + 3 {a}^{2} b + 3a {b}^{2} + {b}^{3} }}$

It is given that (x + 1/x) = 4, Cubing on both sides :

$\sf { {(x + \frac{1}{x} )}^{3} } = {4}^{3}$

Expanding by the formula :

$\sf { {(x + \frac{1}{x} )}^{3} } = {4}^{3} \\ \\ \sf{ {x}^{3} + 3( {x})^{2} ( \frac{1}{x} ) + 3x( { \frac{1}{x} )}^{2} + { (\frac{1}{x} )}^{3} = 64 } \\ \\ \sf{ {x}^{3} + 3x + \frac{3}{x} + { (\frac{1}{x} )}^{3} } = 64 \\ \\ \sf{ {x}^{3} + 3(x + \frac{1}{x}) + { (\frac{1}{x} )}^{3} } = 64 \\ \\ \sf{ {x}^{3} + 3(4) + { (\frac{1}{x} )}^{3} } = 64 \\ \\ \sf{ {x}^{3} + { (\frac{1}{x} )}^{3} } = 64 - 12 \\ \\ \star \: \underline{ \boxed{\sf{ {x}^{3} + { (\frac{1}{x} )}^{3} } = 52}} \: \star$

@SweetestBitter