Question

1. \\(sqrt3{-192m^4n^2}\\)\
2. \\((50x^2y^2)^{\frac{1}{2}}\\)\
3. \\((405a^9b^{10})^{\frac{1}{4}}\\)\
4. \\(sqrt3{81x^2y^3}\\)

Explanation:

Step1: Factor radicand for 15)

$\sqrt[3]{-64 \cdot 3 \cdot m^3 \cdot m \cdot n^2}$

Step2: Simplify cube root for 15)

$\sqrt[3]{-64m^3} \cdot \sqrt[3]{3mn^2} = -4m\sqrt[3]{3mn^2}$

Step3: Rewrite exponent as root for 16)

$\sqrt{50x^2y^2}$

Step4: Simplify square root for 16)

$\sqrt{25 \cdot 2 \cdot x^2y^2} = 5|xy|\sqrt{2}$

Step5: Factor base for 17)

$(81 \cdot 5 \cdot a^8 \cdot a \cdot b^8 \cdot b^2)^{\frac{1}{4}}$

Step6: Simplify 4th root for 17)

$(81a^8b^8)^{\frac{1}{4}} \cdot (5ab^2)^{\frac{1}{4}} = 3a^2b^2\sqrt[4]{5ab^2}$

Step7: Factor radicand for 18)

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

Step8: Simplify cube root for 18)

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

Answer:

1. $-4m\sqrt[3]{3mn^2}$
2. $5|xy|\sqrt{2}$
3. $3a^2b^2\sqrt[4]{5ab^2}$
4. $3y\sqrt[3]{3x^2}$