If 3m + 5n = 0 then what is the value of *m*3+5*n*3*m*3−5*n*3 ?

We are given the equation \( 3m + 5n = 0 \). We need to find the value of the expression \( m^3 + 5n^3 \times m^3 - 5n^3 \).

First, let's simplify the given equation \( 3m + 5n = 0 \). Solving for \( m \), we get:

\[
3m = -5n
\]
\[
m = -\frac{5n}{3}
\]

Now, substitute \( m = -\frac{5n}{3} \) into the expression \( m^3 + 5n^3 \times m^3 - 5n^3 \).

1. Compute \( m^3 \):
\[
m^3 = \left(-\frac{5n}{3}\right)^3 = -\frac{125n^3}{27}
\]

2. Compute \( 5n^3 \times m^3 \):
\[
5n^3 \times m^3 = 5n^3 \times \left(-\frac{125n^3}{27}\right) = -\frac{625n^6}{27}
\]

3. Now, the original expression can be rewritten as:
\[
m^3 + 5n^3 \times m^3 - 5n^3
\]

Substitute the values:
\[
-\frac{125n^3}{27} + \left(-\frac{625n^6}{27}\right) - 5n^3
\]

However, this expression is already simplified. Since the expression involves a cubic equation and is not reducible to simple terms without more specific values for \( n \), we can see that the general form does not lead to any further cancellations.

Therefore, the simplified value of the expression is:

\[
\boxed{-\frac{125n^3}{27} + \left(-\frac{625n^6}{27}\right) - 5n^3}
\]

Let me know if you need more details or have any questions. Here are five related questions you might find interesting:

1. How do you solve systems of linear equations?
2. What are some applications of cubic equations in real-world problems?
3. Can we factorize cubic polynomials?
4. How can you solve higher-degree polynomials in general?
5. What are some techniques for simplifying algebraic expressions?

**Tip:** When solving algebraic equations, always look for common factors and consider simplifying the equation as much as possible before proceeding with substitutions.

Learn how to solve the cubic equation 3m + 5n = 0 and find the value of m^3 + 5n^3 × m^3 - 5n^3. This problem involves linear equations and cubic equations, demonstrating advanced algebraic techniques. Suitable for students in Grades 10-12.

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