https://file.aiquickdraw.com/mathbot/file/1726083414552xqgw3dgo.jpg

The image shows a square root expression with variables. The expression seems to be:

\[
\sqrt{20a^{10}b^6c^9}
\]

To simplify this:

1. **Break down each part**:
   - \( \sqrt{20} \)
   - \( \sqrt{a^{10}} \)
   - \( \sqrt{b^6} \)
   - \( \sqrt{c^9} \)

2. **Simplify the square roots of the variables**:
   - \( \sqrt{a^{10}} = a^5 \) (since \( \sqrt{a^{10}} = a^{10/2} = a^5 \))
   - \( \sqrt{b^6} = b^3 \) (since \( \sqrt{b^6} = b^{6/2} = b^3 \))
   - \( \sqrt{c^9} = c^4\sqrt{c} \) (since \( \sqrt{c^9} = c^{9/2} = c^4\sqrt{c} \))

3. **Simplify \( \sqrt{20} \)**:
   - \( \sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5} \)

4. **Final Simplification**:
   \[
   \sqrt{20a^{10}b^6c^9} = 2a^5b^3c^4\sqrt{5c}
   \]

Thus, the simplified expression is:

\[
2a^5b^3c^4\sqrt{5c}
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you simplify expressions with higher powers under square roots?
2. Can you explain how to handle fractional exponents in square roots?
3. How do you deal with cube roots instead of square roots in a similar expression?
4. What happens when the radicand contains negative numbers?
5. How can this simplification be applied to real-world problems?

### Tip:
When simplifying square roots, always look for perfect squares and simplify variables by dividing their exponents by 2.

Learn how to simplify the square root of the expression 20a^10b^6c^9 step-by-step, using algebraic rules and exponent properties. This problem is suitable for high school students in Grades 9-11.

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