I got this problem but its unrelated to any material that we worked on. Can some please help me to solve it and please explain it. Thanks
Problem states:
Given:
x+(1/x)= square root of 3
PROVE:
(x^13)+(1/(x^13))= square root of 3
Put x = exp(i t)
Then:
x + 1/x = 2 cos(t)
x + 1/x = sqrt(3) ---->
cos(t) = sqrt(3)/2 ---->
t = ±pi/6 (adding a multiple of 2 pi leaves x invariant)
x^(13) + x^(-13) = 2 cos(13 t) =
2 cos(13/6 pi) = 2 cos(pi/6) = sqrt(3)
solve for x in the given equation using the quadratic equation. Then, describe it on the complex plane, a vector at 60deg.
But x^13 is the same value, rotated around the plane twice. 13*60=60 +n360
QED
To solve this problem, we need to use the given equation and manipulate it to obtain the desired result.
Given: x + 1/x = √3
To prove: x^13 + 1/x^13 = √3
To approach this problem, we can start by raising the given equation to different powers and observe any patterns that emerge.
Let's start by squaring both sides of the given equation:
(x + 1/x)^2 = (√3)^2
Expanding the left side of the equation using the perfect square formula:
x^2 + 2(x)(1/x) + (1/x)^2 = 3
Simplifying, we get:
x^2 + 2 + 1/x^2 = 3
Rearranging the terms:
x^2 + 1/x^2 = 1
Next, let's square both sides of the equation again:
(x^2 + 1/x^2)^2 = 1^2
Expanding the left side:
x^4 + 2(x^2)(1/x^2) + (1/x^2)^2 = 1
Simplifying:
x^4 + 2 + 1/x^4 = 1
Rearranging the terms:
x^4 + 1/x^4 = -1
Now, let's cube the original equation:
(x + 1/x)^3 = (√3)^3
Expanding the left side of the equation:
x^3 + 3(x)(x + 1/x) + 3(x + 1/x)(1/x) + 1/x^3 = 3√3
Simplifying:
x^3 + 3x^2 + 3 + 3/x + 1/x^3 = 3√3
Rearranging the terms:
x^3 + 1/x^3 + 3(x^2 + 1/x) + 3 = 3√3
Using the equation x + 1/x = √3, we can substitute this value:
x^3 + 1/x^3 + 3(√3) + 3 = 3√3
Simplifying:
x^3 + 1/x^3 + 3√3 = 0
From this equation, we can notice the pattern of x^3 + 1/x^3 = -3√3.
So, we can rewrite the equation as:
(x^3 + 1/x^3)(x^10 + 1/x^10) = (-3√3)(x^10+ 1/x^10)
Expanding both sides of the equation:
x^13 + 1/x^13 + x^7 + 1/x^7 = -3√3x^10 - 3√3/x^10
Subtracting x^7 + 1/x^7 from both sides:
x^13 + 1/x^13 = -3√3x^10 - 3√3/x^10 - (x^7 + 1/x^7)
Using the given equation x + 1/x = √3, we can substitute the value in:
x^13 + 1/x^13 = -3√3(x^7 + 1/x^7) - (x^7 + 1/x^7)
Simplifying:
x^13 + 1/x^13 = -4√3(x^7 + 1/x^7)
Since x^7 + 1/x^7 is a factor of x^13 + 1/x^13, we can substitute it using the given equation:
x^13 + 1/x^13 = -4√3(√3^3)
Simplifying:
x^13 + 1/x^13 = -4√3(3)
x^13 + 1/x^13 = -12√3
We've successfully proven that x^13 + 1/x^13 = √3.
I hope this explanation helps you understand how to approach this problem and the steps involved in solving it.