Question

if x+1/3=3 calculate x⁴+1/x⁴

Answer 1

Given Equation is x + 1/x = 3.

On squaring both sides, we get

(x + 1/x) = (3)^2

x^2 + 1/x^2 + 2 * x * 1/x = 9

x^2 + 1/x^2 + 2 = 9

x^2 + 1/x^2 = 9 - 2

x^2 + 1/x^2 = 7.


Now,

on cubing both sides, we get

(x + 1/x)^3 = (3)^3

x^3 + 1/x^3 + 3 *x * 1/x(x + 1/x) = 27

x^3 + 1/x^3 + 3 * (3) = 27

x^3 + 1/x^3 + 9 = 27

x^3 + 1/x^3 = 27 - 9

x^3 + 1/x^3 = 18.


Now,

We got x^2 + 1/x^2 = 7

on squaring both sides, we get

(x^2 + 1/x^2)^2 = (7)^2

x^4 + 1/x^4 + 2 * x^2 * 1/x^2 = 49

x^2 + 1/x^2 + 2 = 49

x^4 + 1/x^4 = 49 - 2

x^4 + 1/x^4 = 47.



Hope this helps!

Answer 2

$x + \frac{1}{x} = 3 \\ \\ ( {x + \frac{1}{x} })^{2} = 9 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } + 2 = 9 \\ \\ {x}^{2} + \frac{1}{ {x}^{2} } = 7 \\ \\ ( { {x}^{2} + \frac{1}{ {x}^{2} } })^{2} = 49 \\ \\ {x}^{4} + \frac{1}{ {x}^{4} } + 2 = 49 \\ \\ {x}^{4} + \frac{1}{ {x}^{4} } = 47$