Begin by factoring the integer:

$\displaystyle \sqrt[4]{81x^6y^5z^{10}}$

$\displaystyle \sqrt[4]{3^4x^6y^5z^{10}}$

$\displaystyle 3\sqrt[4]{x^6y^5z^{10}}$

Now, simplify the exponents:

$\displaystyle 3\sqrt[4]{x^4x^2y^4yz^{4}z^4z^2}$

$\displaystyle 3xyzz\sqrt[4]{x^2yz^2}$

$\displaystyle 3xyz^2\sqrt[4]{x^2yz^2}$

Begin by converting the radical into exponent form:

$\displaystyle \sqrt[3]{(3n-2)^3}$

$\displaystyle (3n-2)^3^{(\frac{1}{3})}$

Now, multiply the exponents:

$\displaystyle 3n-2$

Begin by converting the radical into exponent form:

$\displaystyle \sqrt[4]{18a^3bc^5}\cdot \sqrt[4]{27a^2b^6c}$

$\displaystyle (18a^3bc^5)^{\frac{1}{4}}\cdot ({27a^2b^6c})^{\frac{1}{4}}$

Now, combine the bases:

$\displaystyle (486a^5b^7c^6)^{\frac{1}{4}}$

Simplify the integer:

$\displaystyle (2\cdot 3\cdot 3^4a^5b^7c^6)^{\frac{1}{4}}$

Now, simplify the exponents:

$\displaystyle (2\cdot 3\cdot 3^4aa^4b^3b^4c^2c^4)^{\frac{1}{4}}$

Convert back into radical form and simplify:

$\displaystyle \sqrt[4]{2\cdot 3\cdot 3^4aa^4b^3b^4c^2c^4}$

$\displaystyle 3abc\sqrt[4]{2\cdot 3ab^3c^2}$

$\displaystyle 3abc\sqrt[4]{6ab^3c^2}$

Begin by multiplying the numerator and denominator by the complement of the denominator:

$\displaystyle \frac{3-2\sqrt{3}}{2\sqrt{3}+3} \cdot \frac{2\sqrt{3}-3}{2\sqrt{3}-3}$

Use the FOIL method to multiply the radicals. F (first) O (outer) I (inner) L (last)

$\displaystyle \frac{6\sqrt{3}-9-12+6\sqrt{3}}{12-6\sqrt{3}+6\sqrt{3}-9}$

Now, combine like terms:

$\displaystyle \frac{12\sqrt{3}-21}{3}$

Simplify:

$\displaystyle 4\sqrt{3}-7$

Begin by factoring the radicals:

$\displaystyle \sqrt{\frac{1}{5}}-2\sqrt{20}+3\sqrt{5}$

$\displaystyle \sqrt{\frac{1}{5}}-4\sqrt{5}+3\sqrt{5}$

Combine like terms:

$\displaystyle \sqrt{\frac{1}{5}}-\sqrt{5}$

Multiply the left side by $\displaystyle \frac{\sqrt{5}}{\sqrt{5}}$ and the right side by $\displaystyle \frac{5}{5}$

$\displaystyle \frac{1}{\sqrt{5}}-\sqrt{5}$

$\displaystyle \frac{1}{\sqrt{5}}\cdot \frac{\sqrt{5}}{\sqrt{5}}-\sqrt{5}\cdot\frac{5}{5}$

$\displaystyle \frac{\sqrt{5}}{5}-\frac{5\sqrt{5}}{5}$

$\displaystyle \frac{-4\sqrt{5}}{5}$

Begin by converting the radicals into exponent form:

$\displaystyle \sqrt[4]{24a^2b^3c^5}\cdot\sqrt[4]{48a^3b^3c^6}$

$\displaystyle (24a^2b^3c^5)^{\frac{1}{4}}\cdot(48a^3b^3c^6)^{\frac{1}{4}}$

$\displaystyle (3\cdot 2^3a^2b^3c^5)^{\frac{1}{4}}\cdot(3\cdot 2^4a^3b^3c^6)^{\frac{1}{4}}$

Now, combine the bases:

$\displaystyle (3^2\cdot 2^7a^5b^6c^{11})^{\frac{1}{4}}$

$\displaystyle (3^2\cdot 2^7aa^4b^2b^4c^3c^{8})^{\frac{1}{4}}$

Convert back into radical form and simplify:

$\displaystyle \sqrt[4]{3^2\cdot 2^7aa^4b^2b^4c^3c^{8}}$

$\displaystyle 2abc^2\sqrt[4]{3^22^3ab^2c^3}$

$\displaystyle 2abc^2\sqrt[4]{72ab^2c^3}$

Begin by simplifying the right side of the rational expression:

$\displaystyle \sqrt{3mn}+\sqrt{27m^3n}$

$\displaystyle \sqrt{3mn}+3m\sqrt{3mn}$

$\displaystyle 1\sqrt{3mn}+3m\sqrt{3mn}$

Now, combine like terms:

$\displaystyle (1+3m)\sqrt{3mn}$

Begin by using the FOIL method to multiply the radical expression. F (first) O (outer) I (inner) L (last)

$\displaystyle (4-\sqrt{3})(6-\sqrt{3})$

$\displaystyle 24-4\sqrt{3}-6\sqrt{3}+3$

Now, combine like terms:

$\displaystyle 27-10\sqrt{3}$

Begin by multiplying the numerator and denominator by $\displaystyle \frac{\sqrt[3]{(5n)^2}}{\sqrt[3]{(5n)^2}}$:

$\displaystyle \sqrt[3]{\frac{2}{5n}}$

$\displaystyle \frac{\sqrt[3]{2}}{\sqrt[3]{5n}}\cdot$$\displaystyle \frac{\sqrt[3]{(5n)^2}}{\sqrt[3]{(5n)^2}}$

$\displaystyle \frac{\sqrt[3]{50n^2}}{5n}$

The expression cannot be further simplified.

Begin by multiplying the numerator and denominator by the complement of the denominator:

$\displaystyle \frac{2-\sqrt{2}}{4+2\sqrt{2}}$

$\displaystyle \frac{2-\sqrt{2}}{4+2\sqrt{2}} \cdot \frac{4-2\sqrt{2}}{4-2\sqrt{2}}$

$\displaystyle \frac{8-4\sqrt{2}-4\sqrt{2}+4}{16-8}$

Combine like terms and simplify:

$\displaystyle \frac{12-8\sqrt{2}}{8}$

$\displaystyle \frac{3-2\sqrt{2}}{2}$