To find the product of the given four terms, we need to multiply them together step by step.
The given terms are:
[tex]\[
-\frac{1}{2} x^2, \; -\frac{3}{5} xy, \; \frac{2}{3} yz, \; \text{and} \; \frac{5}{7} xyz
\][/tex]
First, let's multiply the coefficients (the numerical parts) together, and then we will deal with the variables.
The coefficients are:
[tex]\[
-\frac{1}{2}, -\frac{3}{5}, \frac{2}{3}, \; \text{and} \; \frac{5}{7}
\][/tex]
Multiplying these coefficients together:
1. Multiply the first two:
[tex]\[
-\frac{1}{2} \times -\frac{3}{5} = \frac{3}{10}
\][/tex]
2. Multiply the result with the third coefficient:
[tex]\[
\frac{3}{10} \times \frac{2}{3} = \frac{2}{10} = \frac{1}{5}
\][/tex]
3. Multiply the result with the fourth coefficient:
[tex]\[
\frac{1}{5} \times \frac{5}{7} = \frac{1}{7}
\][/tex]
Next, let's consider the variables.
From the given terms:
[tex]\[
-\frac{1}{2} x^2 \times -\frac{3}{5} xy \times \frac{2}{3} yz \times \frac{5}{7} xyz
\][/tex]
Combining the variables:
- [tex]\(x^2 \times x \times x \times x = x^5\)[/tex]
- [tex]\(y \times y \times y = y^3\)[/tex]
- [tex]\(z \times z = z^2\)[/tex]
Putting everything together:
[tex]\[
-\frac{1}{2} x^2 \times -\frac{3}{5} xy \times \frac{2}{3} yz \times \frac{5}{7} xyz = \left( \frac{1}{7} \right) x^5 y^3 z^2
\][/tex]
So, the product of the given terms is:
[tex]\[
\boxed{0.14285714285714285 x^5 y^3 z^2 }
\][/tex]
