Given:
x+(1/x)= square root of 3
PROVE:
(x^13)+(1/(x^13))= square root of 3
what does this have to do with physics?
Given:
x+(1/x)= square root of 3
PROVE:
(x^13)+(1/(x^13))= square root of 3
To prove the given statement, we need to show that if x + (1/x) = √3, then (x^13) + (1/(x^13)) = √3.
Let's start by manipulating the expression x + (1/x) = √3. We can rewrite it as:
x + (1/x) = √3
To simplify the equation, we can square both sides:
(x + (1/x))^2 = (√3)^2
Expanding both sides:
x^2 + 2(x)(1/x) + (1/x)^2 = 3
Simplifying further:
x^2 + 2(1) + 1/x^2 = 3
Next, let's subtract 2 from both sides:
x^2 + 1/x^2 = 3 - 2
x^2 + 1/x^2 = 1
Now, we can square this equation again to get rid of the fraction:
(x^2 + 1/x^2)^2 = 1^2
Expanding both sides:
x^4 + 2(x^2)(1/x^2) + 1/x^4 = 1
Simplifying:
x^4 + 2 + 1/x^4 = 1
Subtracting 2 from both sides:
x^4 + 1/x^4 = 1 - 2
x^4 + 1/x^4 = -1
Finally, we can multiply this equation by (x^9) to obtain the expression we need to prove:
(x^4)(x^9) + (1/x^4)(x^9) = (-1)(x^9)
Simplifying:
(x^13) + (1/x^13) = -x^9
Now, let's substitute the value of x + (1/x) from the given information into this equation:
(x^13) + (1/x^13) = - (√3)^9
(x^13) + (1/x^13) = -3^9
(x^13) + (1/x^13) = -19683
Therefore, we can conclude that (x^13) + (1/x^13) = -19683, which is not equal to the square root of 3. Hence, the given statement cannot be proven true.