1. **State the problem:** Given $x=2^{1/2}$, find the value of $$x^2 - \frac{1}{x^2}$$ and discuss if rationalizing the denominator is necessary. 2. **Recall the formula:** We want to evaluate $$x^2 - \frac{1}{x^2}$$. 3. **Calculate $x^2$:** Since $x=2^{1/2}$, then $$x^2 = (2^{1/2})^2 = 2.$$ 4. **Calculate $\frac{1}{x^2}$:** Using the value of $x^2$, $$\frac{1}{x^2} = \frac{1}{2}.$$ 5. **Substitute and simplify:** $$x^2 - \frac{1}{x^2} = 2 - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2}.$$ 6. **Regarding rationalizing the denominator:** Rationalizing the denominator is a technique used to eliminate radicals from the denominator of a fraction. Here, since the denominator is $x^2 = 2$, which is already rational, rationalizing is not necessary. The answer is correct without rationalizing. **Final answer:** $$x^2 - \frac{1}{x^2} = \frac{3}{2}.$$