2. is the positive square root of k , for positive values of k . The following rules apply: Let’s apply these rules to simplifying some expressions. Surds

3. Surds Simplify Simplify Combine the square roots. Example

4. Surds Simplify Simplify Applies to the product of any number of terms. Example

5. Surds Simplify Example

6. Surds Simplify Simplify Factorise. Factorise. Example Express each term in terms of

7. Surds Simplify Expand. Collect terms. Example

8. Surds Simplify Rationalise the denominator. Equivalent form Equivalent to multiplying by 1. Example

9. Surds Simplify Equivalent to multiplying by 1. Difference of two squares. Alternate form of the answer. This will rationalise the denominator. Example

10. Surds, Indices and Logarithms Simplify Equivalent to multiplying by 1. Rationalise the denominator. Example

11. Surds, Indices and Logarithms Solve the equation Check Square both sides of the equation. This clears the surds. Example

12. Surds, Indices and Logarithms Summary 1: Rationalising the Denominator You can swap the ‘ –’ and the ‘+’ signs. If the denominator contains Multiply by If the denominator contains Multiply by

13. Surds Solve the equation Check Rearrange the equation and square both sides. This clears the surds. Solve the equation. A check is always needed due to squaring. Example

14. If we square both sides of an equation then the following can happen: So, solving the equation may give us a different solution. Obviously both cannot be correct. Surds . . ,

15. Another rule to apply to the equality of surds Let’s apply this rule to solving an equation. Surds

16. Find the values of a and b . Equating rational and irrational terms. Solve like solving a pair of simultaneous equations. Expand LHS. Surds Example Solution