1. **Statement of the problem:** Solve for $x$ in the equation $352 = 8.16 \times 2^{x \times 2}$. 2. **Formula and rules:** We will isolate the exponential term and then use logarithms to solve for $x$. 3. **Isolate the exponential:** Divide both sides by 8.16: $$\frac{352}{8.16} = 2^{2x}$$ Calculate the left side: $$43.1372549 \approx 2^{2x}$$ 4. **Apply logarithm base 2:** $$\log_2(43.1372549) = 2x$$ 5. **Calculate the logarithm:** $$2x = \log_2(43.1372549) \approx 5.43$$ 6. **Solve for $x$:** $$x = \frac{5.43}{2} = 2.715$$ **Final answer:** $$x \approx 2.715$$