1. Simplify $\sqrt{12}$. We know $12 = 4 \times 3$ and $\sqrt{4} = 2$. So, $\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$. 2. Simplify $\sqrt{98}$. Since $98 = 49 \times 2$ and $\sqrt{49} = 7$, $\sqrt{98} = \sqrt{49 \times 2} = 7\sqrt{2}$. 3. Simplify $\sqrt{363}$. Factor 363: $363 = 121 \times 3$ and $\sqrt{121} = 11$. So, $\sqrt{363} = 11\sqrt{3}$. 4. Simplify $\sqrt{180}$. $180 = 36 \times 5$ and $\sqrt{36} = 6$. Thus, $\sqrt{180} = 6\sqrt{5}$. 5. Simplify $\sqrt{250}$. $250 = 25 \times 10$ and $\sqrt{25} = 5$. So, $\sqrt{250} = 5\sqrt{10}$. 6. Simplify $\sqrt[4]{48}$. $48 = 16 \times 3$ and $\sqrt[4]{16} = 2$ because $2^4=16$. Therefore, $\sqrt[4]{48} = 2\sqrt[4]{3}$. 7. Simplify $\sqrt[5]{486}$. $486 = 243 \times 2$ and $243 = 3^5$. So, $\sqrt[5]{486} = \sqrt[5]{3^5 \times 2} = 3\sqrt[5]{2}$. 8. Simplify $\sqrt{4x^2y^3}$. $\sqrt{4} = 2$, $\sqrt{x^2} = x$ (since variables positive), and $\sqrt{y^3} = y\sqrt{y}$. So, $\sqrt{4x^2y^3} = 2x y \sqrt{y}$. 9. Simplify $\sqrt{289a^4b^2}$. $\sqrt{289} = 17$, $\sqrt{a^4} = a^2$, $\sqrt{b^2} = b$. Thus, $\sqrt{289a^4b^2} = 17 a^2 b$. 10. Simplify $\sqrt{54x^3y^6}$. $54 = 9 \times 6$, $\sqrt{9} = 3$. $\sqrt{x^3} = x \sqrt{x}$, $\sqrt{y^6} = y^3$. So, $\sqrt{54x^3y^6} = 3 y^3 x \sqrt{6x}$. 11. Simplify $\sqrt{200ab^2}$. $200 = 100 \times 2$, $\sqrt{100} = 10$. $\sqrt{a} = \sqrt{a}$, $\sqrt{b^2} = b$. So, $\sqrt{200ab^2} = 10 b \sqrt{2a}$. 12. Simplify $\sqrt{64a^6b}$. $\sqrt{64} = 8$, $\sqrt{a^6} = a^3$, $\sqrt{b} = \sqrt{b}$. Thus, $\sqrt{64a^6b} = 8 a^3 \sqrt{b}$. 13. Simplify $\sqrt[3]{1000x^6y^5}$. $1000 = 10^3$, so $\sqrt[3]{1000} = 10$. $\sqrt[3]{x^6} = x^{6/3} = x^2$. $\sqrt[3]{y^5} = y^{1} \sqrt[3]{y^2} = y \sqrt[3]{y^2}$. So, $\sqrt[3]{1000x^6y^5} = 10 x^2 y \sqrt[3]{y^2}$. 14. Simplify $\sqrt[4]{162x^2y^4z^6}$. $162 = 81 \times 2$, $\sqrt[4]{81} = 3$ because $3^4=81$. $\sqrt[4]{x^2} = x^{1/2} = \sqrt{x}$. $\sqrt[4]{y^4} = y$. $\sqrt[4]{z^6} = z^{6/4} = z^{3/2} = z \sqrt{z}$. So, $\sqrt[4]{162x^2y^4z^6} = 3 y z \sqrt{x} \sqrt{z} \sqrt[4]{2}$. 15. Simplify $\sqrt[3]{128x^5y^4z^3}$. $128 = 64 \times 2$, $\sqrt[3]{64} = 4$. $\sqrt[3]{x^5} = x^{1} \sqrt[3]{x^2} = x \sqrt[3]{x^2}$. $\sqrt[3]{y^4} = y \sqrt[3]{y}$. $\sqrt[3]{z^3} = z$. So, $\sqrt[3]{128x^5y^4z^3} = 4 x y z \sqrt[3]{2 x^2 y}$. Final answers: 1. $2\sqrt{3}$ 2. $7\sqrt{2}$ 3. $11\sqrt{3}$ 4. $6\sqrt{5}$ 5. $5\sqrt{10}$ 6. $2\sqrt[4]{3}$ 7. $3\sqrt[5]{2}$ 8. $2 x y \sqrt{y}$ 9. $17 a^2 b$ 10. $3 x y^3 \sqrt{6 x}$ 11. $10 b \sqrt{2 a}$ 12. $8 a^3 \sqrt{b}$ 13. $10 x^2 y \sqrt[3]{y^2}$ 14. $3 y z \sqrt{x} \sqrt{z} \sqrt[4]{2}$ 15. $4 x y z \sqrt[3]{2 x^2 y}$