Operations with Radicals GSE Honors Algebra II Keeper 19

Adding or Subtracting Radical Expressions **You can only add or subtract radical expressions if they have common radicands and indexes.**

Steps for Adding or Subtracting Radical Expressions Step 1: Simplify all radicals completely. Step 2: Combine the outside terms of common radicands. The radicand does NOT change.

Example 1 2 5 +7 5

Example 2 3 4 2 −5 2

Example 3 3 3 +6 27

Example 4 3 4 48 − 4 243

Example 5 −5 12 +9 3 40 +6 48 −4 3 135

Multiplying Radical Expressions **You can only multiply radicals with the same index.**

Steps for Multiplying Radical Expressions Step 1: Multiply whatever is “outside” of the radical. Step 2: Multiply whatever is on the “inside” of the radical. Step 3: Simplify the radical completely.

Example 6 4 2 ∙3 10

Example 7 2 3 10 𝑎 2 𝑏 ∙4 3 5𝑎𝑏 𝑐 2

Example 8 2 3 14 − 7

Example 9 2 3 6 +5 3 4 2

Check for Understanding 2 3 40 + 3 135 3 45 𝑥 7 −2𝑥 20 𝑥 5 −2 45 −3 20 −2 6 −3 6 3 −2 6 192 − 6 320 3 3 ∙ 3 9 4 𝑥 2 3 9 𝑥 4 𝑦 ∙2 3 6 𝑥 2 𝑦 3 2𝑥 18𝑥𝑦 ∙5𝑥 6 𝑥 5 𝑦 3 3 4−3 5 2 3 4 3 10 −6 3 81 4 5 15 2 5 16 +7 5 50

Dividing Radical Expressions Step 1: Use the Quotient Property of Radicals to rewrite the expression. This property allows you to write a fraction under one big radical or the numerator and denominator under separate radicals. 2𝑥 5 = 2𝑥 5 Step 2: Simplify the numerator. Step 3: Simplify the denominator. You MUST remember that you can NEVER leave a radical in the denominator!

Example 10 2 𝑥 2 9 𝑦 2 What if the radical in the denominator does NOT eliminate when we simplify?

You must RATIONALIZE THE DENOMINATOR! *Think about how many of the number/variable we have in the denominator and how many more you need to match the index. Let's call this "just enough." Step 1: Multiply the numerator and denominator by "just enough" to eliminate the radical in the denominator. Step 2: Simplify the radical in the numerator. Step 3: Simplify any outside terms.

Example 11 4 20

Example 12 20𝑥 50 𝑥 2

Special Cases: When there is a radical being added or subtracted in the denominator, we must multiply by the RADICAL CONJUGATE in order to rationalize the denominator. 3+ 2 → 4− 5 →

Example 13 5 2− 7

Example 14 4 −2+3 5

Example 15 3 𝑥 2𝑦 2

Check for Understanding 8 100 3 7 3 𝑏 4 𝑎 5𝑥 3 10𝑥 𝑦 3 2 9− 2 4 𝑥 2 𝑦 3 5 3 2𝑥 𝑦 2