Question

x+1/x=✓5.then find the value of x3+1/x3

Answer 1

Heya friend,
Here is the answer you were looking for:
$x + \frac{1}{x} = \sqrt{5}$

On Cubing both the sides we get,

${(x + \frac{1}{x}) }^{3} = {( \sqrt{5} )}^{3}$

Using the identity :

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

${(x)}^{3} + {( \frac{1}{x}) }^{3} + 3 \times x \times \frac{1}{x} (x + \frac{1}{x} ) = 5 \sqrt{5} \\ \\ {x}^{3} + \frac{1}{ {x}^{3} } + 3( \sqrt{5} ) = 5 \sqrt{5} \\ \\ {x}^{3} + \frac{1}{ {x}^{3} } + 3 \sqrt{5} = 5 \sqrt{5} \\ \\ {x}^{3} + \frac{1}{ {x}^{3} } = 5 \sqrt{5} - 3 \sqrt{5} \\ \\ \\ {x}^{3} + \frac{1}{ {x}^{3} } = 2 \sqrt{5}$

Hope this helps!!!

Feel free to ask in comment section if you have any doubt regarding to my answer...

@Mahak24

Thanks...
☺☺

Answer 2

Hey.......!!! here is ur answer........☺️☺️☺️

Given that,

x+1/x = √5........(1)

On squaring of equation (1) both the sides...

(x+1/x)² = 5

=>x²+1/x²+2x(1/x) = 5

=>x²+1/x²+2 = 5

=>x²+1/x² = 5–2 = 3............(2)

Now we have to find....... x³+1/x³

Then, x³+1/x³ = (x+1/x)(x²–x.1/x+1/x²)

{ we know that a³+b³ = (a+b)(a²–ab+b²) }

=>x³+1/x³ = (x+1/x).(x²+1/x²–1)

from equation (1) and (2)

=>x³+1/x³ = √5.(3–1)

=>x³+1/x³ = 2√5 ««««««ANS.

I hope it will help you.........✌️✌️✌️