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 \(\sqrt[6]{\frac{1}{8}}\).