https://r2.erweima.ai/mathbot/image/1726184414690m94tfa3s.jpg

Let's solve the given roots step by step.

1. \( \sqrt{4a^2b^4} \)
   - Simplifying each part:
   \[
   \sqrt{4} = 2, \quad \sqrt{a^2} = a, \quad \sqrt{b^4} = b^2
   \]
   - Final answer:
   \[
   2ab^2
   \]

2. \( \sqrt{25x^6y^7} \)
   - Simplifying each part:
   \[
   \sqrt{25} = 5, \quad \sqrt{x^6} = x^3, \quad \sqrt{y^7} = y^3\sqrt{y}
   \]
   - Final answer:
   \[
   5x^3y^3\sqrt{y}
   \]

3. \( \sqrt[3]{27a^4b^6} \)
   - Simplifying each part:
   \[
   \sqrt[3]{27} = 3, \quad \sqrt[3]{a^4} = a^{4/3}, \quad \sqrt[3]{b^6} = b^2
   \]
   - Final answer:
   \[
   3a^{4/3}b^2
   \]

4. \( \sqrt[3]{-8a^3b^6c^{10}} \)
   - Simplifying each part:
   \[
   \sqrt[3]{-8} = -2, \quad \sqrt[3]{a^3} = a, \quad \sqrt[3]{b^6} = b^2, \quad \sqrt[3]{c^{10}} = c^{10/3}
   \]
   - Final answer:
   \[
   -2ab^2c^{10/3}
   \]

5. \( \sqrt{64x^8y^9} \)
   - Simplifying each part:
   \[
   \sqrt{64} = 8, \quad \sqrt{x^8} = x^4, \quad \sqrt{y^9} = y^4\sqrt{y}
   \]
   - Final answer:
   \[
   8x^4y^4\sqrt{y}
   \]

6. \( \sqrt[4]{16a^8b^{15}} \)
   - Simplifying each part:
   \[
   \sqrt[4]{16} = 2, \quad \sqrt[4]{a^8} = a^2, \quad \sqrt[4]{b^{15}} = b^{15/4}
   \]
   - Final answer:
   \[
   2a^2b^{15/4}
   \]

7. \( \sqrt[5]{x^{15}y^{20}z^{26}} \)
   - Simplifying each part:
   \[
   \sqrt[5]{x^{15}} = x^3, \quad \sqrt[5]{y^{20}} = y^4, \quad \sqrt[5]{z^{26}} = z^{26/5}
   \]
   - Final answer:
   \[
   x^3y^4z^{26/5}
   \]

8. \( \sqrt[3]{-64a^3x^5y^{18}} \)
   - Simplifying each part:
   \[
   \sqrt[3]{-64} = -4, \quad \sqrt[3]{a^3} = a, \quad \sqrt[3]{x^5} = x^{5/3}, \quad \sqrt[3]{y^{18}} = y^6
   \]
   - Final answer:
   \[
   -4ax^{5/3}y^6
   \]

9. \( \sqrt[5]{-243m^5n^{18}} \)
   - Simplifying each part:
   \[
   \sqrt[5]{-243} = -3, \quad \sqrt[5]{m^5} = m, \quad \sqrt[5]{n^{18}} = n^{18/5}
   \]
   - Final answer:
   \[
   -3mn^{18/5}
   \]

10. \( \sqrt[3]{1000x^{10}y^{13}} \)
    - Simplifying each part:
    \[
    \sqrt[3]{1000} = 10, \quad \sqrt[3]{x^{10}} = x^{10/3}, \quad \sqrt[3]{y^{13}} = y^{13/3}
    \]
    - Final answer:
    \[
    10x^{10/3}y^{13/3}
    \]

Would you like further explanations for any step? Here are five related questions to help reinforce understanding:

1. How do you simplify fractional exponents?
2. What is the difference between square roots and cube roots?
3. How can you rewrite a radical as a fractional exponent?
4. What is the general rule for solving \( n \)-th roots of variables with exponents?
5. How do negative numbers affect the solution when finding cube roots?

**Tip:** Always break down radicals by separating constants and variables for easier simplification.

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Learn how to simplify expressions involving square roots, cube roots, and other radical forms. This problem set includes algebraic expressions with variables raised to various powers, perfect squares, and higher-degree roots. Suitable for students in Grades 9-11.

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