1. The problem is to evaluate the expression $\frac{36.6 - 21.5382}{\frac{\sqrt{2}}{2}}$. 2. The formula used here is a simple division of the difference of two numbers by a fraction. 3. First, calculate the numerator: $36.6 - 21.5382 = 15.0618$. 4. The denominator is $\frac{\sqrt{2}}{2}$. Recall that dividing by a fraction is the same as multiplying by its reciprocal. 5. So, the expression becomes: $$\frac{15.0618}{\frac{\sqrt{2}}{2}} = 15.0618 \times \frac{2}{\sqrt{2}}$$ 6. Simplify the multiplication: $$15.0618 \times \frac{2}{\sqrt{2}} = 15.0618 \times \frac{2}{\sqrt{2}} = 15.0618 \times \frac{2}{\sqrt{2}}$$ 7. Multiply numerator and denominator inside the fraction by $\sqrt{2}$ to rationalize: $$\frac{2}{\sqrt{2}} = \frac{2 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2}$$ 8. So the expression simplifies to: $$15.0618 \times \sqrt{2}$$ 9. Using $\sqrt{2} \approx 1.4142$, calculate: $$15.0618 \times 1.4142 \approx 21.3$$ 10. Therefore, the value of the expression is approximately $21.3$.