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The given expression is:

\[
\frac{3}{4} m^3 - \frac{m^2 n}{4} \left( 2 - \frac{n^2}{5} \left( 40 (mn)^{-1} - 5 m^{-2} \right) \right)
\]

Let me break it down and simplify it step by step.

### Step 1: Expand the parentheses

Start with the term inside the parentheses:

\[
\left( 2 - \frac{n^2}{5} \left( 40 (mn)^{-1} - 5 m^{-2} \right) \right)
\]

First, simplify the inner part:
\[
40 (mn)^{-1} = \frac{40}{mn}
\]
\[
5 m^{-2} = \frac{5}{m^2}
\]
So, the term inside becomes:
\[
2 - \frac{n^2}{5} \left( \frac{40}{mn} - \frac{5}{m^2} \right)
\]

### Step 2: Simplify the term

Now expand the outer multiplication by \(\frac{n^2}{5}\):

\[
\frac{n^2}{5} \times \frac{40}{mn} = \frac{8n}{m}
\]
\[
\frac{n^2}{5} \times \frac{5}{m^2} = \frac{n^2}{m^2}
\]

Thus, the term becomes:
\[
2 - \left( \frac{8n}{m} - \frac{n^2}{m^2} \right)
\]

Now simplify:
\[
2 - \frac{8n}{m} + \frac{n^2}{m^2}
\]

### Step 3: Multiply by \(-\frac{m^2 n}{4}\)

Now, take this expression and multiply it by \(-\frac{m^2 n}{4}\):

\[
-\frac{m^2 n}{4} \left( 2 - \frac{8n}{m} + \frac{n^2}{m^2} \right)
\]

Multiply each term inside:
\[
-\frac{m^2 n}{4} \times 2 = -\frac{m^2 n}{2}
\]
\[
-\frac{m^2 n}{4} \times \left( -\frac{8n}{m} \right) = +\frac{2 m n^2}{1}
\]
\[
-\frac{m^2 n}{4} \times \frac{n^2}{m^2} = -\frac{n^3}{4}
\]

So, this part becomes:
\[
-\frac{m^2 n}{2} + 2mn^2 - \frac{n^3}{4}
\]

### Step 4: Combine with the first term

Now, combine the first term \(\frac{3}{4} m^3\) with the expanded result:

\[
\frac{3}{4} m^3 - \left( \frac{m^2 n}{2} - 2mn^2 + \frac{n^3}{4} \right)
\]

This expression is the simplified form of the given equation.

Would you like further details or clarifications on any part of the process?

Here are 5 related questions for practice:
1. How do you simplify expressions involving exponents and fractions?
2. What are some rules for expanding parentheses in algebraic expressions?
3. How do you factor complex polynomial expressions?
4. How can you use substitution to simplify algebraic terms?
5. What is the process for solving rational expressions involving multiple variables?

**Tip:** Always keep track of negative signs when expanding expressions, as they can easily change the outcome of your result.

Simplify the algebraic expression: (3/4)m^3 - (m^2 * n)/4(2 - (n^2/5)(40(mn)^-1 - 5m^-2))

Explore how to simplify the algebraic expression involving exponents and fractions. This step-by-step breakdown includes key algebraic concepts such as exponent laws, fraction multiplication, and polynomial expansion. Ideal for high school students in Grades 10-12.

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