This document discusses three cases of rationalizing radicals, or removing radicals from denominators. Case 1 is when the radical is a square root, which is removed by multiplying the numerator and denominator by the square root. Case 2 is when the radical is not a square root, requiring multiplying by a power of the radical. Case 3 is when the denominator has a sum or subtraction of square roots, handled by multiplying numerator and denominator by the conjugate of the denominator. Exercises are provided for each case.

1. Rationalising radicals It is not usual to leave a expression with surds or radicals in the denominator. Rationalising a radical expression is “removing” the radical in the denominator. We are going to do this in 3 different cases:

2. Rationalising radicals Case 1 : The radical in the denominator is a square root. For example: We multiply the numerator and the denominator by that square root. In the example:

3. Rationalising radicals Case 1 :

4. Rationalising radicals Case 1 : Try this exercise now:

5. Rationalising radicals Case 1 : Try this exercise now:

6. Rationalising radicals Case 2 : The radical in the denominator is not a square root. For example:

7. Rationalising radicals Case 2 : The radical in the denominator is not a square root. For example: This a little tricky to solve. We multiply the numerator and the denominator by a power of the radical: the “necessary” power to remove the radical. In the example:

8. Rationalising radicals Case 2 :

9. Rationalising radicals Case 2 : Try this exercise now:

10. Rationalising radicals Case 2 : Try this exercise now:

11. Rationalising radicals Case 3 : The denominator is a sum or subtraction with square roots. For example:

12. Rationalising radicals Case 3 : The denominator is a sum or subtraction with square roots. For example: We multiply the numerator and denominator by the same expression in the denominator but changing the sign.

13. Rationalising radicals Case 3 :

14. Rationalising radicals Case 3 : Try this one now:

15. Rationalising radicals Case 3 : Try this one now:

16. Rationalising radicals New term!!! Conjugate radicals We can say that in the last 2 examples, we multiplied both the numerator and the denominator by the conjugate of the denominator.