To expand the expression $(2m^{-n^2})(3m^2 - 5n^2)$, we will use the distributive property (also known as the FOIL method for binomials).

The expression has two parts:

1. Distribute $2m^{-n^2}$ to both terms inside the parentheses: $3m^2$ and $-5n^2$.

Step-by-step expansion:

$(2m^{-n^2})(3m^2 - 5n^2) = (2m^{-n^2})(3m^2) - (2m^{-n^2})(5n^2)$

1. First term: $(2m^{-n^2})(3m^2) = 6m^{-n^2 + 2}$ (Using the rule $m^a \cdot m^b = m^{a + b}$)

2. Second term: $-(2m^{-n^2})(5n^2) = -10m^{-n^2}n^2$

Final expanded form:

$6m^{-n^2 + 2} - 10m^{-n^2}n^2$

This is the fully expanded version of the expression.

Do you need more details or have any questions about the steps?

Here are 5 related questions:

1. How does the distributive property work in algebraic expressions?
2. What are the rules for simplifying expressions involving exponents?
3. How do you combine terms with the same base but different exponents?
4. How do negative exponents affect the simplification of terms?
5. Can this approach be applied to polynomials with more terms?

Tip: When multiplying terms with exponents, always add the exponents if the bases are the same.