The expression in the image is:

$\sqrt[3]{64x^{10}y^3}$

To simplify this expression, we can follow these steps:

Step 1: Break Down the Radicand

The radicand (the expression under the cube root) is $64x^{10}y^3$. We'll simplify each part individually.

• $64$ is a perfect cube, because $4^3 = 64$.
• $x^{10}$ can be rewritten as $x^9 \times x^1$, where $x^9$ is a cube ($(x^3)^3$).
• $y^3$ is already a perfect cube.

Step 2: Apply the Cube Root

The cube root of a product is the product of the cube roots:

[ \sqrt[3]{64x^{10}y^3} = \sqrt[3]{64} \times \sqrt[3]{x^{10}} \times \sqrt[3]{y^3} ]

Step 3: Simplify Each Part

• $\sqrt[3]{64} = 4$
• $\sqrt[3]{x^{10}} = x^{10/3} = x^3 \times x^{1/3}$ (separating out the perfect cube part)
• $\sqrt[3]{y^3} = y$

Step 4: Combine the Results

Now we combine everything:

$4x^3y \times \sqrt[3]{x}$

This is the simplified form of the expression:

$4x^3y \sqrt[3]{x}$

Let me know if you need further clarifications or more details!

Follow-up Questions:

1. What is the cube root of $x^{9}$ and how does it relate to cube roots in general?
2. How do you simplify cube roots of higher powers of variables?
3. How would this expression change if the root were a square root instead of a cube root?
4. What happens if $x$ and $y$ are negative? How would the simplification process differ?
5. Can you apply a similar method to higher-degree roots, such as the fourth or fifth root?

Tip:

When simplifying expressions with cube roots, always look for perfect cubes to simplify first!