Question

IF X∧2+1/X∧2=7 , FIND THE VALUE OF X∧3+1/X∧3






PLS ANSWER ITS AN EMERGENCY

Answer 1

Hi !

So,

${x}^{2} + \frac{1}{ {x}^{2} } = 7$

We can write it as,

${x}^{2} + \frac{1}{ {x}^{2} } = 9 - 2 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } + 2 = 9$

We know that,

${a}^{2} + {b}^{2} + 2ab = {(a + b)}^{2}$

Assuming a = x
and b = 1/x

We can write,

${(x + \frac{1}{x} )}^{2} = {(3)}^{2} \\ \\ x + \frac{1}{x} = 3$

Cubing both the sides we get,

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

Using the identity :

${(a + b)}^{3} = {a}^{3} + {b}^{3} + 3ab(a + b)$

${(x)}^{3} + {( \frac{1}{x} )}^{3} + 3 \times x \times \frac{1}{x} (x + \frac{1}{x} ) = 27\\ \\ {x}^{3} + \frac{1}{ {x}^{3} } + 3(3) = 27 \\ \\ {x}^{3} + \frac{1}{ {x}^{3} } + 9 = 27 \\ \\ {x}^{3} + \frac{1}{ {x}^{3} } = 27 - 9 \\ \\ {x}^{3} + \frac{1}{{x}^{3} } = 18$

Any doubt arises, please ask ^-^