The expression \( x^3 - \frac{1}{125} \) can be factored using the difference of cubes formula. The difference of cubes formula is:

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

In this case, we can rewrite \(\frac{1}{125}\) as \( \left(\frac{1}{5}\right)^3 \). Thus, the expression becomes:

\[
x^3 - \left(\frac{1}{5}\right)^3
\]

Using the difference of cubes formula:

\[
x^3 - \left(\frac{1}{5}\right)^3 = \left( x - \frac{1}{5} \right)\left( x^2 + x\left(\frac{1}{5}\right) + \left(\frac{1}{5}\right)^2 \right)
\]

Simplifying further:

\[
x^3 - \frac{1}{125} = \left( x - \frac{1}{5} \right)\left( x^2 + \frac{1}{5}x + \frac{1}{25} \right)
\]

This is the factorized form of the expression \( x^3 - \frac{1}{125} \).

Would you like more details or have any questions?

Here are 8 related questions you might find interesting:

1. How do you factor expressions using the sum of cubes formula?
2. What are some other examples of the difference of cubes?
3. How can you use the difference of squares to factor expressions?
4. What is the process for simplifying expressions with cube roots?
5. How do you solve equations involving cube roots?
6. Can the difference of cubes formula be used in solving cubic equations?
7. How does the difference of cubes formula apply to polynomial division?
8. What is the general method to factor a cubic polynomial?

**Tip:** Remember, the difference of cubes formula is a powerful tool for simplifying and solving cubic expressions. It's worth mastering!

Learn how to factorize the expression x^3 - 1/125 using the difference of cubes formula step-by-step. This problem involves understanding algebraic concepts like polynomials and cube roots. Suitable for Grades 10-12 students.

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