Question

1. \\(\frac{3}{\sqrt{6}}\cdot\frac{\sqrt{6}}{\sqrt{0}}\\) \\(\frac{3\sqrt{0}}{30}\\) 12. \\(\sqrt3{\frac{5}{3x}}\\) \\(\frac{\sqrt3{5}}{\sqrt3{3x}}\cdot\frac{\sqrt3{9x^2}}{\sqrt3{9x^2}}\\) \\(\frac{\sqrt3{45x^2}}{3x}\\) 13. \\(\sqrt3{\frac{18x^2}{24x}}\\) \\(\frac{\sqrt3{3x}}{\sqrt3{4}}\cdot\frac{\sqrt3{2}}{\sqrt3{2}}\\) \\(\frac{\sqrt3{6x}}{2}\\) 14. \\(\sqrt3{\frac{x^2}{9y^2}}\\) \\(\frac{\sqrt3{x^2}}{\sqrt3{9y^2}}\cdot\frac{\sqrt3{3y}}{\sqrt3{3y}}\\) \\(\frac{\sqrt3{3x^2y}}{3y}\\) 15. \\(\sqrt3{3x^2}\cdot\sqrt3{x^2}\cdot\sqrt3{9x^3}\\) \\(\sqrt3{3x^4}\cdot\sqrt3{9x^3}\\) \\(\sqrt9{24x^{12}}\\) \\(\sqrt3{24}x^4\\) \\(\sqrt3{8}\sqrt3{3}\\) \\(2x^4\sqrt3{3}\\) 16. \\(\sqrt{x^5y^5}\cdot3\sqrt{2x^7y^6}\\) \\(3\sqrt{2x^{12}y^{11}}\\) \\(3x^6\sqrt{2y^{11}}\\)

Explanation:

Step1: Simplify Q11: Multiply fractions

$\frac{3}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{10}} = \frac{3\sqrt{6}}{\sqrt{60}}$

Step2: Simplify Q11: Rationalize denominator

$\frac{3\sqrt{6}}{\sqrt{60}} = \frac{3\sqrt{6}}{\sqrt{4 \cdot 15}} = \frac{3\sqrt{6}}{2\sqrt{15}} = \frac{3\sqrt{90}}{2 \cdot 15} = \frac{3 \cdot 3\sqrt{10}}{30} = \frac{9\sqrt{10}}{30} = \frac{3\sqrt{10}}{10}$

Step3: Simplify Q12: Rationalize denominator

$\sqrt[3]{\frac{5}{3x}} = \frac{\sqrt[3]{5}}{\sqrt[3]{3x}} \cdot \frac{\sqrt[3]{9x^2}}{\sqrt[3]{9x^2}} = \frac{\sqrt[3]{45x^2}}{\sqrt[3]{27x^3}} = \frac{\sqrt[3]{45x^2}}{3x}$

Step4: Simplify Q13: Reduce fraction inside root

$\sqrt[3]{\frac{18x^2}{24x}} = \sqrt[3]{\frac{3x}{4}}$

Step5: Simplify Q13: Rationalize denominator

$\sqrt[3]{\frac{3x}{4}} = \frac{\sqrt[3]{3x}}{\sqrt[3]{4}} \cdot \frac{\sqrt[3]{2}}{\sqrt[3]{2}} = \frac{\sqrt[3]{6x}}{2}$

Step6: Simplify Q14: Rationalize denominator

$\sqrt[3]{\frac{x^2}{9y^2}} = \frac{\sqrt[3]{x^2}}{\sqrt[3]{9y^2}} \cdot \frac{\sqrt[3]{3y}}{\sqrt[3]{3y}} = \frac{\sqrt[3]{3x^2y}}{\sqrt[3]{27y^3}} = \frac{\sqrt[3]{3x^2y}}{3y}$

Step7: Simplify Q15: Combine radicals

$\sqrt[3]{3x^2} \cdot \sqrt[3]{x^2} \cdot \sqrt[3]{9x^3} = \sqrt[3]{3x^2 \cdot x^2 \cdot 9x^3} = \sqrt[3]{27x^7}$

Step8: Simplify Q15: Simplify radical

$\sqrt[3]{27x^7} = \sqrt[3]{27x^6 \cdot x} = 3x^2\sqrt[3]{x}$

Step9: Simplify Q16: Combine radicals

$\sqrt{x^5y^5} \cdot 3\sqrt{2x^7y^6} = 3\sqrt{2x^{12}y^{11}}$

Step10: Simplify Q16: Simplify radical

$3\sqrt{2x^{12}y^{11}} = 3x^6y^5\sqrt{2y}$

Answer:

1. $\frac{3\sqrt{10}}{10}$
2. $\frac{\sqrt[3]{45x^2}}{3x}$
3. $\frac{\sqrt[3]{6x}}{2}$
4. $\frac{\sqrt[3]{3x^2y}}{3y}$
5. $3x^2\sqrt[3]{x}$
6. $3x^6y^5\sqrt{2y}$