If x^2 + (1/x^2) =7, find the exact value of x^5 + (1/x^5).?

Question

Exact meaning no approximations.

Answer 1

lets us denote x^n + 1/x^n as x(n)

we need to find x(5) given x(2) =7

if x(2) = y then we have y+1/y = 7 or y^2-7y+1=0 and therefore
y=(7 +/- sqr(45))/2 from which we can find x and then find x^5....

that is one way...

another way is x(1) ^ 2 = x(2) + 2 and hence x(1) = 3

so x(1) ^ 5 = x(5) + (5 x^4/x + 10 x^3/x^2 + 10 x^2/x^3 + 5 x/x^4)
= x(5) + 5 x(3) + 10 x(1)

but x(1)^3 = x(3) + 3 x(1) or 27 = x(3) + 9 or x(3)=18

hence 3^5 = 243 = x(5) + 5*18 + 10 * 3

or x(5) = 243 - 90 - 30 = 123

Answer 2

OK... so x^2 + (1/x^2) = 7.
Multiply both sides by x squared, and you get:
x^4 + 1 = 7x^2
Rearrange so you can use the quadratic equation:
x^4 - 7x^2 + 1 = 0

Using the quadratic equation, you get:
x^2 = (7 plus minus square root of 45) / 2
Therefore, x = sqrt(((7+sqrt(45))/2)

Note that x can't be sqrt(((7 - sqrt(45))/2), since that would be the square root of a negative number.

Once you have that value of x, just plug it into the x^5 + (1/x^5) formula and simplify.

Hopefully that helps!

Answer 3

Let u = x^2

u + 1/u = 7

Solve for u - solve for x and substitute into other expression.

There will be two exact values for the expression because there are 2 real roots for the first equation.