https://file.aiquickdraw.com/mathbot/file/1733235547744e54g3vmq.jpg

Let's analyze the given mathematical expression in the image:

The expression inside the cube root is:

\[
\sqrt[3]{\frac{64x^3y^3}{8z^3}}
\]

### Step-by-step Simplification:

1. **Simplify constants:**
   - \( \frac{64}{8} = 8 \).

   So, the expression becomes:
   \[
   \sqrt[3]{8 \cdot \frac{x^3y^3}{z^3}}
   \]

2. **Simplify the variables:**
   - \( \frac{x^3y^3}{z^3} \) remains as is since all powers are 3.

   The expression now is:
   \[
   \sqrt[3]{8x^3y^3z^{-3}}
   \]

3. **Split the cube root:**
   - The cube root can be split as:
   \[
   \sqrt[3]{8} \cdot \sqrt[3]{x^3} \cdot \sqrt[3]{y^3} \cdot \sqrt[3]{z^{-3}}
   \]

4. **Simplify each term:**
   - \( \sqrt[3]{8} = 2 \) (since \( 2^3 = 8 \)),
   - \( \sqrt[3]{x^3} = x \),
   - \( \sqrt[3]{y^3} = y \),
   - \( \sqrt[3]{z^{-3}} = \frac{1}{z} \).

   So the expression simplifies to:
   \[
   2 \cdot x \cdot y \cdot \frac{1}{z}
   \]

5. **Combine terms:**
   - This results in:
   \[
   \frac{2xy}{z}
   \]

### Final Answer:
\[
\frac{2xy}{z}
\]

Let me know if you want further explanation or details!

---

**Relative Questions:**

1. How can cube roots be applied to real-world problems?
2. What is the difference between square roots and cube roots in terms of their applications?
3. Why is \( \sqrt[3]{z^{-3}} \) simplified to \( \frac{1}{z} \)?
4. How does simplifying a fraction help in solving cube root expressions?
5. What happens if we change the exponents from cubes to other powers?

**Tip:** Always simplify constants first in a fraction under a root; this often reduces the complexity of the entire expression.

Simplify the cube root expression: \( \sqrt[3]{\frac{64x^3y^3}{8z^3}} \).

Explore how to simplify the cube root expression \( \sqrt[3]{\frac{64x^3y^3}{8z^3}} \) using properties of cube roots and exponent rules. Includes a step-by-step breakdown for Grades 8-10 students.

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