Question

X - 1 / X=√5 and x2+1/x2=7 find the value of x3+1/x3​

Answer 1

Answer:

24 is the correct answers

Hope you understood the answer

Answer 2

★Given:-

• $\sf x - \dfrac{1}{x} = \sqrt{5}$

• $\sf x^2 + \dfrac{1}{x^2} = 7$

★To find→

• $\sf x^3 + \dfrac{1}{x^3}$

★Solution:-

First we will find $\sf x + \dfrac{1}{x}$

We know that,

(a+b)² = a²+b²+2ab

Using this identity,

• $\sf \leadsto\Bigg (x + \dfrac{1}{x} \Bigg )^2 = (x)^2 + \Bigg ( \dfrac{1}{x} \Bigg )^2 + 2 \times x \times \dfrac{1}{x}$

• $\sf \leadsto\Bigg (x + \dfrac{1}{x} \Bigg )^2 = x^2 + \dfrac{1}{x^2}+ 2$

• $\sf \leadsto \Bigg (x + \dfrac{1}{x} \Bigg )^2 = 7 + 2$

• $\sf \leadsto\Bigg (x + \dfrac{1}{x} \Bigg )^2 = 9$

• $\sf \leadsto x + \dfrac{1}{x} = \sqrt{9}$

• $\sf \leadsto x + \dfrac{1}{x} = 3$

_________________________

Now, we will find $\sf x^3+\dfrac{1}{x^3}$

We know that,

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

Using this identity,

$\sf \longmapsto x^3 + \dfrac{1}{x^3} = \Bigg ( x + \dfrac{1}{x} \Bigg ) \Bigg ( (x)^2 + \Bigg ( \dfrac{1}{x} \Bigg )^2 - x \times \dfrac{1}{x} \Bigg )$

$\sf \longmapsto x^3 + \dfrac{1}{x^3} = (3 )(x^2 +\dfrac{1}{x^2} - 1)$

$\sf \longmapsto x^3 + \dfrac{1}{x^3} = (3 )(7 - 1)$

$\sf \longmapsto x^3 + \dfrac{1}{x^3} = (3 )(6)$

$\sf \longmapsto x^3 + \dfrac{1}{x^3} = 18$

$\boxed{ \bold{x^3 + \dfrac{1}{x^3} = 18}}$