Theory:

The four basic operations on surds are:
• Addition and subtraction of surds
• Multiplication and division of surds
(i) Addition and subtraction of surds: The similar or like surds can be added or subtracted using the property:

$$a \sqrt[n]{b} \pm c \sqrt[n]{b} = (a \pm c) \sqrt[n]{b}$$, where $$b > 0$$.

(ii) Multiplication and division of surds: The similar or like surds can be multiplied or divided using the following properties:

Multiplication properties:

1. $$\sqrt[n]{a} \times \sqrt[n]{b} = \sqrt[n]{ab}$$

2. $$a \sqrt[n]{b} \times c \sqrt[n]{d} = ac \sqrt[n]{bd}$$ where $$b, d > 0$$

Division properties:

1. $$\frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$$

2. $$\frac{a \sqrt[n]{b}}{c \sqrt[n]{d}} = \frac{a}{c} \sqrt[n]{\frac{b}{d}}$$ where $$b, d > 0$$
Example:
1. Subtract $$15 \sqrt{11}$$ from $$29 \sqrt{11}$$.

Solution:

$$29 \sqrt{11} - 15 \sqrt{11} = (29 - 15) \sqrt{11} = 14 \sqrt{11}$$

Therefore, the solution is $$14 \sqrt{11}$$.

2. Simplify $$12 \sqrt[3]{192} - 5 \sqrt[3]{2187} + 10 \sqrt[3]{648}$$.

Solution:

$$12 \sqrt[3]{192} - 5 \sqrt[3]{2187} + 10 \sqrt[3]{648} = 12 \sqrt[3]{64 \times 3} - 5 \sqrt[3]{729 \times 3} + 10 \sqrt[3]{216 \times 3}$$

$$= 12 \sqrt[3]{4^3 \times 3} - 5 \sqrt[3]{9^3 \times 3} + 10 \sqrt[3]{6^3 \times 3}$$

$$= 12(4 \sqrt[3]{3}) - 5(9 \sqrt[3]{3}) + 10(6 \sqrt[3]{3})$$

$$= 48 \sqrt[3]{3} - 45 \sqrt[3]{3} + 60 \sqrt[3]{3}$$

$$= (48 - 45 + 60)\sqrt[3]{3}$$

$$= 63 \sqrt[3]{3}$$

Therefore, the solution is $$63 \sqrt[3]{3}$$.

3. Multiply $$\sqrt{80}$$ and $$4 \sqrt{45}$$

Solution:

$$\sqrt{80} \times 4 \sqrt{45} = \sqrt{4 \times 4 \times 5} \times \sqrt{3 \times 3 \times 5}$$

$$= (4 \times \sqrt{5}) \times (3 \times \sqrt{5}$$

$$= (4 \times 3)(\sqrt{5} \times \sqrt{5})$$

$$= 12 \sqrt{25}$$

Therefore, the solution is $$12 \sqrt{25}$$.

4. Divide $$\sqrt[6]{2}$$ by $$\sqrt[3]{4}$$

Solution:

$$\frac{\sqrt[6]{2}}{\sqrt[3]{4}} = \frac{2^{\frac{1}{6}}}{4^{\frac{1}{3}}}$$

Take LCM of denominators $$6$$ and $$3$$, we have:

$$= \frac{2^{\frac{1}{6}}}{4^{\frac{2}{6}}}$$

$$= (\frac{2}{4^2})^{\frac{1}{6}}$$

$$= (\frac{2}{4 \times 4})^{\frac{1}{6}}$$

$$= (\frac{1}{8})^{\frac{1}{6}}$$

$$= \sqrt[6]{\frac{1}{8}}$$

Therefore, the solution is $$\sqrt[6]{\frac{1}{8}}$$.