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The given problem asks to multiply and leave the response as individual summands. Let's work step by step to simplify and expand the expressions.

The equation provided is:
\[
m^{a + \frac{1}{2}} - 2m^a \left( \frac{5}{2} m^{\frac{2}{3}} + \frac{1}{2} m^{a + \frac{3}{5}} - \frac{3}{2} m^{3a + 4} \right)
\]
and 
\[
m^{a - \frac{3}{2}} - m^{a - \frac{1}{3}} + m^{a - \frac{2}{7}}.
\]

### Step 1: Expanding the expression
We need to expand the terms inside the first expression.

1. \( m^{a + \frac{1}{2}} \) is left as is for now.
2. \( -2m^a \times \frac{5}{2} m^{\frac{2}{3}} = -5 m^{a + \frac{2}{3}} \)
3. \( -2m^a \times \frac{1}{2} m^{a + \frac{3}{5}} = -m^{2a + \frac{3}{5}} \)
4. \( -2m^a \times \left(-\frac{3}{2} m^{3a + 4}\right) = 3m^{4a + 4} \)

Thus, the expanded first part becomes:
\[
m^{a + \frac{1}{2}} - 5m^{a + \frac{2}{3}} - m^{2a + \frac{3}{5}} + 3m^{4a + 4}.
\]

### Step 2: Simplifying the second part
The second expression is:
\[
m^{a - \frac{3}{2}} - m^{a - \frac{1}{3}} + m^{a - \frac{2}{7}}.
\]

These are already individual summands, so they remain unchanged.

### Final Answer:
Combining both expressions:
\[
m^{a + \frac{1}{2}} - 5m^{a + \frac{2}{3}} - m^{2a + \frac{3}{5}} + 3m^{4a + 4} \quad \text{and} \quad m^{a - \frac{3}{2}} - m^{a - \frac{1}{3}} + m^{a - \frac{2}{7}}.
\]

Would you like more details on any of the steps or additional explanation?

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Here are 5 related questions:
1. How do you multiply terms with the same base but different exponents?
2. What rules govern the simplification of exponents in expressions like this?
3. Can you further simplify any of the terms in this expanded expression?
4. How does factoring affect expressions with exponents?
5. How do you handle negative exponents when simplifying expressions?

**Tip:** When multiplying terms with the same base, you add their exponents.

Multiplique y deje su respuesta en sumandos individuales:

m^(a + 1/2) - 2m^a(5/2 m^(2/3) + 1/2 m^(a + 3/5) - 3/2 m^(3a + 4)) y m^(a - 3/2) - m^(a - 1/3) + m^(a - 2/7)

Learn how to expand and simplify exponential expressions using algebraic rules. This problem involves exponent rules and the distributive property, and is suitable for high school students in Grades 10-12.

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