How accurate is the solution to this surd question.

Question

This is a SURD problem, I totally understand step one but not the remaining steps. My question is how did step 2 (as seen in the picture) come about?

Answer 1

First, it should be clear that if $x = 3 + \sqrt{8}$ and $1/x = 3 - \sqrt{8}$, we have $x + 1/x = 6$.

I think the question is more about what ist the intuition behind considering $x + 1/x$ when the question asks for $x^4 + 1/x^4$. The key observation is here that if you use the binomial formula, you get

$$\left ( x + \frac{1}{x} \right )^2 = x^2 + \frac{1}{x^2} + 2 \cdot x \cdot \frac{1}{x} = x^2 + \frac{1}{x^2} + 2$$

and therefore, the expression $x^2 + 1/x^2$ can be easily calculated by

$$x^2 + \frac{1}{x^2} = \left ( x + \frac{1}{x} \right )^2 - 2.$$

Since you know the value of $x + 1/x$, the right hand side in the above equation is easily obtained. Now we if have this, a similar argument using the binomial formula lets one comfortably conclude the value of $x^4 + 1/x^4$.