https://file.aiquickdraw.com/mathbot/file/1725812392679h456a2o6.jpg

The expression in the image is:

\[
\sqrt[6]{64x^{10}y^3}
\]

To simplify this, we'll follow these steps:

### Step 1: Simplify the numbers
The number 64 is a perfect power of 2, since:
\[
64 = 2^6
\]
So we can rewrite the expression as:
\[
\sqrt[6]{(2^6)(x^{10})(y^3)}
\]

### Step 2: Apply the 6th root
The 6th root can now be distributed across the terms:
\[
\sqrt[6]{2^6} \cdot \sqrt[6]{x^{10}} \cdot \sqrt[6]{y^3}
\]

- \(\sqrt[6]{2^6} = 2\), since the 6th root of \(2^6\) is just 2.
- For the \(x\)-term: \(\sqrt[6]{x^{10}} = x^{10/6} = x^{5/3}\).
- For the \(y\)-term: \(\sqrt[6]{y^3} = y^{3/6} = y^{1/2}\).

### Step 3: Combine the results
Now, combining the terms, the simplified expression becomes:
\[
2x^{5/3}y^{1/2}
\]

So, the simplified expression is:

\[
2x^{5/3}y^{1/2}
\]

Let me know if you'd like more details or have any further questions!

### Related Questions:
1. How do you simplify radical expressions with variables?
2. What is the process for simplifying expressions with fractional exponents?
3. Can you explain how roots and powers are related in algebra?
4. How would you simplify a cube root of an expression like \(x^9y^6\)?
5. What are the rules for simplifying expressions involving both powers and roots?

### Tip:
When simplifying expressions with exponents, always check if the base is a perfect power to help reduce roots efficiently.

Simplify the expression: \( \sqrt[6]{64x^{10}y^3} \)

Learn how to simplify the expression \( \sqrt[6]{64x^{10}y^3} \) step by step. This problem introduces key concepts in radicals, exponentiation, and simplification. Suitable for high school students in Grades 10-12.

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