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The given expression is:

\[
\left( -\frac{5}{2}x^2y \right) \left( \frac{3}{25} xy^2 \right) \left( -\frac{10}{3} x^3 \right) \left( -\frac{3}{4}x^2 y \right)
\]

### Step 1: Combine the constants (numerators and denominators)
The constants are:

\[
\left( -\frac{5}{2} \right), \left( \frac{3}{25} \right), \left( -\frac{10}{3} \right), \left( -\frac{3}{4} \right)
\]

Multiply these constants together:

\[
-\frac{5}{2} \cdot \frac{3}{25} \cdot -\frac{10}{3} \cdot -\frac{3}{4}
\]

### Step 2: Combine the variables
Now, combine the powers of \(x\) and \(y\):

- \(x^2 \cdot x \cdot x^3 \cdot x^2 = x^{2+1+3+2} = x^8\)
- \(y \cdot y^2 \cdot y \cdot 1 = y^{1+2+1} = y^4\)

### Step 3: Simplify the constants

Now simplify the product of the constants:

\[
-\frac{5}{2} \cdot \frac{3}{25} = -\frac{15}{50} = -\frac{3}{10}
\]
\[
-\frac{3}{10} \cdot -\frac{10}{3} = 1
\]
\[
1 \cdot -\frac{3}{4} = -\frac{3}{4}
\]

### Step 4: Final expression

Now combine the simplified constants with the variables:

\[
-\frac{3}{4} x^8 y^4
\]

Thus, the simplified result is:

\[
-\frac{3}{4}x^8 y^4
\]

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Do you need further details on any step or have questions?

### Related Questions:
1. How do you simplify expressions with fractions and exponents?
2. What are the rules for multiplying powers with the same base?
3. How do negative signs affect multiplication?
4. Can you explain how to multiply terms with different variables?
5. How can expressions with large exponents be broken down for easier calculation?

### Tip:
Always group constants, like terms, and exponents before starting the multiplication process in complex algebraic expressions.

Simplify the expression: (-5/2 * x^2 * y)(3/25 * x * y^2)(-10/3 * x^3)(-3/4 * x^2 * y)

This example shows how to simplify a complex polynomial expression involving multiplication of variables with exponents and fractional coefficients. Learn how to apply the rules of exponents and simplify constants for a clean result.

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