1. **State the problem:** Simplify the expression $$\left(\frac{10}{0.77x}\right)^2$$. 2. **Recall the rule:** When you have a fraction raised to a power, you raise both numerator and denominator to that power: $$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$$. 3. **Apply the rule:** $$\left(\frac{10}{0.77x}\right)^2 = \frac{10^2}{(0.77x)^2}$$. 4. **Calculate numerator:** $$10^2 = 100$$. 5. **Calculate denominator:** $$(0.77x)^2 = (0.77)^2 \times x^2 = 0.5929x^2$$. 6. **Write the simplified expression:** $$\frac{100}{0.5929x^2}$$. 7. **Optional: Simplify the fraction by dividing numerator and denominator by 0.5929:** $$\frac{\cancel{100}}{\cancel{0.5929}x^2} = \frac{100/0.5929}{x^2} \approx \frac{168.75}{x^2}$$. **Final answer:** $$\boxed{\frac{168.75}{x^2}}$$