1. Radical Operations Adding & Subtracting Radicals 1Copyright (c) 2011 by Lynda Greene Aguirre

2. Radical Expressions A RADICAL is the symbol best known as a square root symbol. Copyright © 2011 by Lynda Aguirre2 A RADICAL EXPRESSION has radical terms and does not have an equal sign. The object under the radical is called the RADICAND

3. Adding (& Subtracting) Terms with radicals can only be added if their radicands are the same These two terms have the same radicand: “3” 3Copyright (c) 2011 by Lynda Greene Aguirre

4. Addition: Same radicand 1. Add the coefficients 2. Bring down the radical 4Copyright (c) 2011 by Lynda Greene Aguirre

5. Subtraction: Same radicand 1. Subtract the coefficients 2. Bring down the radical 5Copyright (c) 2011 by Lynda Greene Aguirre

6. Addition and Subtraction: Same radicand 1. Add and Subtract the coefficients 2. Bring down the radical 6Copyright (c) 2011 by Lynda Greene Aguirre Note: if there is no number in front of a radical, it is a “1”.

7. Different Radicands Simplify terms with different radicands, then add or subtract their coefficients. Radicands are not the same so we cannot add or subtract these terms. Try to simplify the terms (see “simplify radicals” notes for more details ) 7Copyright (c) 2011 by Lynda Greene Aguirre

8. Simplify the Radicals NOW the radicands are the same so we can add the coefficients 8Copyright (c) 2011 by Lynda Greene Aguirre

9. Different Radicands Rule: We can only add or subtract radicals with the same radicands, so try to simplify them first. 9Copyright (c) 2011 by Lynda Greene Aguirre 9 and 4 are both perfect squares, so we can replace them with their square roots

10. Different Radicands Rule: We can only add or subtract radicals with the same radicand, so here, we can only combine the last 2 terms. 10Copyright (c) 2011 by Lynda Greene Aguirre 7 and 3 are both prime numbers, so we can’t simplify them any further. The “1” in front of the radical can be dropped

11. Different Radicands These radicands cannot be reduced, so there is nothing that can be done to simplify this expression 11Copyright (c) 2011 by Lynda Greene Aguirre Solution

12. Practice: Addition & Subtraction See following slides for the step-by-step solutions 12Copyright (c) 2011 by Lynda Greene Aguirre

13. Practice (key): Addition & Subtraction 13Copyright (c) 2011 by Lynda Greene Aguirre

14. Practice: Addition & Subtraction 14Copyright (c) 2011 by Lynda Greene Aguirre

15. Practice: Addition & Subtraction 15Copyright (c) 2011 by Lynda Greene Aguirre

16. Multiplication of Radicals Copyright (c) 2011 by Lynda Greene Aguirre16

17. Multiplying Radicals Rule: Copyright (c) 2011 by Lynda Greene Aguirre17 Example: Then simplify if possible

18. Multiplying Radicals Rule: Copyright (c) 2011 by Lynda Greene Aguirre18 Example: Then simplify if possible Distribute Radicands are not the same, so this cannot be simplified further

19. Copyright (c) 2011 by Lynda Greene Aguirre19 Multiplication of Radicals

20. Copyright (c) 2011 by Lynda Greene Aguirre20 Multiplication of Radicals

21. Copyright (c) 2011 by Lynda Greene Aguirre21 Another path to the same answer: There are often several correct paths to the answer. Some are shorter than others. Multiplication of Radicals

22. Copyright (c) 2011 by Lynda Greene Aguirre22 Expand the exponent to see the whole problem Process: FOIL Combine Like Terms and Simplify Radicals Multiplication of Radicals

23. Multiplying Radicals Copyright (c) 2011 by Lynda Greene Aguirre23 Multiply using FOIL Add the terms with the same radicand This is the solution

24. Practice Problems Copyright (c) 2011 by Lynda Greene Aguirre24

25. Division of Radicals Copyright (c) 2011 by Lynda Greene Aguirre25

26. Dividing Radicals Copyright (c) 2011 by Lynda Greene Aguirre26 Rules: Outside numbers on the top can be divided by Outside numbers on the bottom. Inside numbers on the top can be divided by Inside numbers on the bottom. Reduce the outside numbers Reduce the inside numbers Radicals are not allowed on the bottom (denominator): see “rationalizing the denominator” notes for more details on this process Short version of this: Multiply top and bottom by the radicand. (This shortcut only works for square roots) Note: These are the same thing and can be changed as needed

27. Dividing Radicals Copyright (c) 2011 by Lynda Greene Aguirre27 Reduce the outside numbers Reduce the inside numbers Rationalize the Denominator Note: see “properties of radicals” notes for this “splitting the radical” property Simplify the radical Reduce the fraction

28. Dividing Radicals Copyright (c) 2011 by Lynda Greene Aguirre28 Rationalize the denominator Simplify the top radical Terms with + or – signs between them cannot be reduced separately DistributeSimplify the radicals This can only be reduced if the coefficients (outside numbers) could all be divided by the same number. Since they can’t, this is the solution

29. Practice Problems Copyright (c) 2011 by Lynda Greene Aguirre29

30. Rationalize Denominator Copyright (c) 2011 by Lynda Greene Aguirre30

31. Rationalize the Denominator Rule: Radicals on the bottom of a fraction must be removed. Copyright (c) 2011 by Lynda Greene Aguirre31 Type 1: Single Term -Multiply the top and bottom by the same radical. Type 2: Binomial (Two Terms) -Multiply the top and bottom by the complex conjugate (same thing, different signs). Note: Don’t leave a negative on the bottom of a fraction. Move it in front of the fraction and/or multiply the top by it (distribute).

32. Root Rationalize the Denominator Higher Order Radicals If the power does not form a perfect number, multiply the top and bottom by enough extra terms so that the powers will add up to a perfect number. Copyright (c) 2011 by Lynda Greene Aguirre32 We only have 2 sevens To take out a radical, we must create a “perfect” number. Recall that this means that the power must be divisible by the root. Root Power we need 1 more to make 3. Use the rational exponent property

33. Rationalize the Denominator Higher Order Radicals If the power does not form a perfect number, multiply the top and bottom by enough extra terms so that the powers will add up to a perfect number. Copyright (c) 2011 by Lynda Greene Aguirre33 We only have 3 twos we need 2 more to make 5. Use the rational exponent property Always check to see if you can reduce (cancel) or simplify radicals when you reach the end of a problem.

34. Copyright (c) 2011 by Lynda Greene Aguirre34 Rationalize the Denominator

35. For free math notes visit our website: www.greenebox.com Copyright (c) 2011 by Lynda Greene Aguirre35