To solve the expression [tex]\( -\frac{3}{2} \pi \sqrt{2} \)[/tex], we follow several steps to understand each component and compute the final result.
1. Understand the components:
- The fraction [tex]\(-\frac{3}{2}\)[/tex] represents a single coefficient.
- [tex]\(\pi\)[/tex] is a constant representing the ratio of the circumference of a circle to its diameter, approximately equal to [tex]\(3.141592653589793\)[/tex].
- [tex]\(\sqrt{2}\)[/tex] is the square root of 2, which is approximately [tex]\(1.4142135623730951\)[/tex].
2. Break down the multiplication:
- First, we multiply [tex]\(\pi\)[/tex] and [tex]\(\sqrt{2}\)[/tex].
- Then, we multiply the result with the coefficient [tex]\(-\frac{3}{2}\)[/tex].
3. Multiply [tex]\(\pi\)[/tex] with [tex]\(\sqrt{2}\)[/tex]:
- Given [tex]\(\pi \approx 3.141592653589793\)[/tex] and [tex]\(\sqrt{2} \approx 1.4142135623730951\)[/tex],
- Multiplying these numbers gives [tex]\( 3.141592653589793 \times 1.4142135623730951 \approx 4.442882938158366 \)[/tex].
4. Include the coefficient [tex]\(-\frac{3}{2}\)[/tex]:
- Now, multiply this intermediate result by [tex]\(-\frac{3}{2}\)[/tex], which is [tex]\(-1.5\)[/tex]:
- [tex]\(-1.5 \times 4.442882938158366 \approx -6.664324407237549\)[/tex].
5. Final Result:
- The final value of the expression [tex]\( -\frac{3}{2} \pi \sqrt{2} \)[/tex] is approximately [tex]\(-6.664324407237549\)[/tex].
So, the step-by-step simplification shows that:
[tex]\[
-\frac{3}{2} \pi \sqrt{2} \approx -6.664324407237549
\][/tex]
