Simplifying Expressions with Surds

Recap of Rules

Multiplying/Dividing Surds:
- Multiply or divide the numbers inside the root.
- Example:
  - ( \sqrt{5} \times \sqrt{6} = \sqrt{5 \times 6} = \sqrt{30} )
  - ( \sqrt{20} \div \sqrt{10} = \sqrt{20/10} = \sqrt{2} )
Adding/Subtracting Surds:
- Cannot add or subtract different surds directly.
- Example:
  - ( \sqrt{13} + \sqrt{6} \neq \sqrt{19} )
- Can add/subtract coefficients of same surds.
- Example:
  - ( 2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3} )
  - ( 6\sqrt{7} - 2\sqrt{7} = 4\sqrt{7} )
Multiplying a Surd by Itself:
- Surd disappears as it becomes a whole number.
- Example:
  - ( \sqrt{7} \times \sqrt{7} = 7 )

Goal: Simplify expressions into the form ( a + b\sqrt{n} ) where ( a ) and ( b ) are integers.

Simplify ( \sqrt{125} ):
- ( \sqrt{125} = \sqrt{5 \times 25} = \sqrt{5} \times 5 = 5\sqrt{5} )
Simplify ( 2\sqrt{45} ):
- ( \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} )
- ( 2\sqrt{45} = 2 \times 3\sqrt{5} = 6\sqrt{5} )
- Remember: ( -6\sqrt{5} )
Expand ((\sqrt{5} + 2)^2):
- ( \sqrt{5} \times \sqrt{5} = 5 )
- ( \sqrt{5} \times 2 = 2\sqrt{5} )
- ( 2 \times \sqrt{5} = 2\sqrt{5} )
- ( 2 \times 2 = 4 )
- Simplified: ( 5 + 4 + 4\sqrt{5} = 9 + 4\sqrt{5} )
Combine Terms:
- ( 5\sqrt{5} - 6\sqrt{5} + 9 + 4\sqrt{5} = 3\sqrt{5} + 9 )
- Rewritten as: ( 9 + 3\sqrt{5} )

Simplify ( \sqrt{48} ):
- ( \sqrt{48} = 4\sqrt{3} )
Simplify ( 2\sqrt{75} ):
- ( 2\sqrt{75} = 10\sqrt{3} )
Simplify ( (\sqrt{3})^2 ):
- ( (\sqrt{3})^2 = 3 )
Combine Terms:
- ( 4\sqrt{3} + 10\sqrt{3} + 3 = 14\sqrt{3} + 3 )
- Rewritten as: ( 3 + 14\sqrt{3} )

Understanding and applying these rules helps in simplifying surds to desired forms.
Useful for exams and mathematical applications.
Share knowledge with peers for collaborative learning.