Question

If x-1/x=2.23,find the values of x^2 +1/x^2,x+1/x,x^3+1/x^3

Answer 1

Step-by-step explanation:

the value of (x^3 + 1/x^2)?

I’ll answer a related question. Prove the following theorem:

Theorem: If [math](x + \frac 1 x) = 4[/math] and [math](x^2 + \frac{1}{x^3}) = 12[/math] then the moon is made of green cheese.

Proof: Suppose, for the purpose of contradiction, that the moon is not made of green cheese. Since [math]x+\frac 1 x = 4,[/math] it follows that [math]x[/math] and [math]\frac 1 x[/math] are roots of the quadratic equation in [math]z,[/math]

[math]\qquad z^2–4z+1=0[/math]

The roots of this equation are [math]2+\sqrt 3[/math] and [math]2-\sqrt 3,[/math] which are [math]x[/math] and [math]\frac 1 x[/math] in some order. So [math]x^2+\frac{1}{x^3}[/math] is not [math]12,[/math] but rather either [math]33-11\sqrt3[/math] or [math]33+11\sqrt3,[/math] a contradiction, which completes the proof.

[math]\,\\[/math]My point? Any statement, true or not, validly follows from a false premise.

x + 1/x = 4 , squaring both sides we get x^2 + 1/x^2= 14, Similarly taking the power of 3 on both sides we get x^3 + 1/x^3 +3(x + 1/x) = 64 which implies x^3 + 1/x^3 = 52.

Now adding up the above two equations, we get

x^2 + 1/x^2 +x^3 + 1/x^3 = 66

it is given that x^2 + 1/x^3 =12 hence substituting this value in the above equation gives the answer of the question

i.e - x^3 + 1/x^2 + 12 = 66 which implies that x^3 + 1/x^2 = 54.

Hence the answer is 54.