To factor the expression \(216m^3 - 343n^6\), we recognize it as a difference of cubes. The general formula for factoring a difference of cubes is:

\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\]

Here, \(216m^3\) can be written as \((6m)^3\) and \(343n^6\) can be written as \((7n^2)^3\). Therefore:

\[
216m^3 - 343n^6 = (6m)^3 - (7n^2)^3
\]

Using the difference of cubes formula:

\[
(6m)^3 - (7n^2)^3 = (6m - 7n^2)\left((6m)^2 + (6m)(7n^2) + (7n^2)^2\right)
\]

Now, let's simplify the terms inside the parentheses:

\[
(6m)^2 = 36m^2
\]
\[
(6m)(7n^2) = 42mn^2
\]
\[
(7n^2)^2 = 49n^4
\]

So, the fully factored form is:

\[
216m^3 - 343n^6 = (6m - 7n^2)(36m^2 + 42mn^2 + 49n^4)
\]

Would you like more details on any part of this process or have any further questions?

Here are some related questions you might find interesting:
1. How do you factor the sum of cubes?
2. Can you factor the expression \(8a^3 + 125b^3\)?
3. What is the general approach to factoring quadratic expressions?
4. How do you solve a cubic equation by factoring?
5. What are some common mistakes when factoring expressions?

**Tip:** When factoring cubes, always check if the expression is a perfect cube and apply the difference or sum of cubes formulas accordingly.

Learn how to factor the expression 216m³ - 343n⁶ by applying the difference of cubes formula. This detailed solution covers key algebraic concepts, simplifying each step to break down the expression into its factored form. Suitable for students in Grades 9-12.

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