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36 4 7 8 3 Radical Operations Adding & Subtracting Radicals Copyright (c) 2011 by Lynda Greene Aguirre 1 Radical Expressions A RADICAL is the symbol best known as a square root symbol. A RADICAL EXPRESSION has radical terms and does not have an equal sign. The object under the radical is called the RADICAND Copyright © 2011 by Lynda Aguirre 2 Adding (& Subtracting) Terms with radicals can only be added if their radicands are the same These two terms have the same radicand: “3” Copyright (c) 2011 by Lynda Greene Aguirre 3 Addition: Same radicand 2. Bring down the radical 1. Add the coefficients Copyright (c) 2011 by Lynda Greene Aguirre 4 Subtraction: Same radicand 2. Bring down the radical 1. Subtract the coefficients Copyright (c) 2011 by Lynda Greene Aguirre 5 Addition and Subtraction: Same radicand 2 5 5 7 5 Note: if there is no number in front of a radical, it is a “1”. 2 5 1 5 7 5 1. Add and Subtract the coefficients ( 2 1 7) 5 2. Bring down the radical 8 5 Copyright (c) 2011 by Lynda Greene Aguirre 6 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) Copyright (c) 2011 by Lynda Greene Aguirre 7 Simplify the Radicals NOW the radicands are the same so we can add the coefficients Copyright (c) 2011 by Lynda Greene Aguirre 8 Different Radicands 4 9 5 4 Rule: We can only add or subtract radicals with the same radicands, so try to simplify them first. 9 and 4 are both perfect squares, so we can replace them with their square roots 9 3 4 2 4(3) 5(2) 12 10 2 Copyright (c) 2011 by Lynda Greene Aguirre 9 Different Radicands 2 7 5 3 4 3 Rule: We can only add or subtract radicals with the same radicand, so here, we can only combine the last 2 terms. 2 7 (5 4) 3 7 and 3 are both prime numbers, so we can’t simplify them any further. 2 7 1 3 The “1” in front of the radical can be dropped 2 7 3 Copyright (c) 2011 by Lynda Greene Aguirre 10 Different Radicands 3 6 2 These radicands cannot be reduced, so there is nothing that can be done to simplify this expression 3 6 2 Solution Copyright (c) 2011 by Lynda Greene Aguirre 11 Practice: Addition & Subtraction See following slides for the step-by-step solutions Copyright (c) 2011 by Lynda Greene Aguirre 12 Practice (key): Addition & Subtraction Copyright (c) 2011 by Lynda Greene Aguirre 13 Practice: Addition & Subtraction Copyright (c) 2011 by Lynda Greene Aguirre 14 Practice: Addition & Subtraction Copyright (c) 2011 by Lynda Greene Aguirre 15 Multiplication of Radicals Copyright (c) 2011 by Lynda Greene Aguirre 16 Multiplying Radicals Rule: Example: Then simplify if possible Copyright (c) 2011 by Lynda Greene Aguirre 17 Multiplying Radicals Rule: Example: Distribute Then simplify if possible Radicands are not the same, so this cannot be simplified further Copyright (c) 2011 by Lynda Greene Aguirre 18 Multiplication of Radicals Copyright (c) 2011 by Lynda Greene Aguirre 19 Multiplication of Radicals Copyright (c) 2011 by Lynda Greene Aguirre 20 Multiplication of Radicals Another path to the same answer: There are often several correct paths to the answer. Some are shorter than others. Copyright (c) 2011 by Lynda Greene Aguirre 21 Multiplication of Radicals Expand the exponent to see the whole problem Process: FOIL Combine Like Terms and Simplify Radicals Copyright (c) 2011 by Lynda Greene Aguirre 22 Multiplying Radicals Multiply using FOIL x 3 x 7 x x 7 x 3x 21 Add the terms with the same radicand ( x 7) x 3x 21 This is the solution Copyright (c) 2011 by Lynda Greene Aguirre 23 Practice Problems 3 6 7 3 2 5 3 x 3x 7 x 2 x 13 2 63 2 2 15 2 x 5 x 3 6x 7 x 7 2 3 x 2x 3 2 Copyright (c) 2011 by Lynda Greene Aguirre 24 Division of Radicals Copyright (c) 2011 by Lynda Greene Aguirre 25 Rules: Dividing Radicals 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 a a b b Note: These are the same thing and can be changed as needed Short version of this: Multiply top and bottom by the radicand. (This shortcut only works for square roots) Copyright (c) 2011 by Lynda Greene Aguirre 26 Dividing Radicals Reduce the outside numbers Rationalize the Denominator Simplify the radical 3 18 9 4 1 9 3 2 9 3 2 2 2 Reduce the inside numbers Note: see “properties of radicals” notes for this “splitting the radical” property Reduce the fraction 18 9 2 3 2 3(2) 6 3 4 Copyright (c) 2011 by Lynda Greene Aguirre 2 2 27 Dividing Radicals 2 18 4 3 Simplify the top radical Terms with + or – signs between them cannot be reduced separately Distribute 2 18 3 4 3 3 2 3 54 Rationalize the denominator Simplify the radicals 4 9 2 3 9 6 2 3 3 6 4(3) 12 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 Copyright (c) 2011 by Lynda Greene Aguirre 28 Practice Problems 2 9 4 3 2 3 5 2 3x 3 3 2 2 3 10 2 3 x 3 3 Copyright (c) 2011 by Lynda Greene Aguirre 29 Rationalize Denominator Copyright (c) 2011 by Lynda Greene Aguirre 30 Rationalize the Denominator Rule: Radicals on the bottom of a fraction must be removed. 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). Copyright (c) 2011 by Lynda Greene Aguirre 31 Rationalize the Denominator Higher Order Radicals 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 Root Root Power Power Power 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. We only have 2 sevens we need 1 more to make 3. Use the rational exponent property Copyright (c) 2011 by Lynda Greene Aguirre 32 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. 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. Copyright (c) 2011 by Lynda Greene Aguirre 33 Rationalize the Denominator Copyright (c) 2011 by Lynda Greene Aguirre 34 For free math notes visit our website: www.greenebox.com Copyright (c) 2011 by Lynda Greene Aguirre 35
