9.1 Properties of Radicals

What We Will Learn Use properties of radicals to simplify Perform operations with radicals

Essential Question How can you multiply and divide square roots?

Needed Vocab Radical expression: expression that contains a radical Simplest form: cannot be broken down any further Like radicals: same index and radicand

Ex. 1 Using Product Property 108 36 × 3 6 3 24 4 × 6 2 6 75 25 × 3 5 3 128 64 × 2 8 2 Steps 1. Find biggest perfect square that divides into number with no decimal 2. Separate 𝑟𝑎𝑑𝑖𝑐𝑎𝑛𝑑 into perfect square and number left after dividing 𝑝𝑒𝑟𝑓𝑒𝑐𝑡 𝑠𝑞𝑢𝑎𝑟𝑒 ∗ 𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 3. Square root perfect square 4. leave 𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 alone

Ex. 1 Continue Simplify Letters 𝑎 6 𝑎 3 𝑎 6 𝑏 12 𝑎 3 𝑏 6 𝑐 11 𝑐 10 × 𝑐 1 𝑐 5 𝑐 𝑑 15 𝑒 9 𝑑 14 × 𝑑 1 × 𝑒 8 × 𝑒 1 𝑑 7 𝑒 4 𝑑𝑒 𝑎 10 𝑏 11 Leave evens alone 𝑎 10 × 𝑏 10 × 𝑏 1 𝑎 5 𝑏 5 𝑏 If exponent is even, then just cut in half If exponent is odd, then find biggest even number Split into the even number exponent and exponent of one Split even number in half, and leave exponent of one alone

Ex. 1 Continued Combining Numbers and Letters 108 𝑥 4 36 × 3 × 𝑥 4 6 𝑥 2 3 120 𝑎 6 𝑏 9 4 × 30 × 𝑎 6 × 𝑏 9 4 × 30 × 𝑎 6 × 𝑏 8 × 𝑏 1 2 𝑎 3 𝑏 4 30𝑏 Your Practice 125 𝑛 7 𝑡 9 25 × 5 × 𝑛 6 × 𝑛 1 × 𝑡 8 × 𝑡 1 5 𝑛 3 𝑡 4 5𝑛𝑡 Steps 1. Split numbers first Use first notes for numbers 2. Split letters second Use notes for letters 3. Do perfect square and simplify letters

Ex. 2 Using Quotient Property 15 64 15 64 15 8 125 144 125 144 125 12 25 × 5 12 5 5 12 Steps 1. Square root top and bottom 2. Simplify using example 1 notes 108 𝑥 5 36𝑦 8 108𝑥 5 36𝑦 8 36 × 3 × 𝑥 4 × 𝑥 1 36 × 𝑦 8 6𝑥 2 3𝑥 6𝑦 4 𝑥 2 3𝑥 𝑦 4

Your Practice 128𝑥 7 64𝑦 6 128𝑥 7 64𝑦 6 64 × 2 × 𝑥 6 × 𝑥 1 64 × 𝑦 6 128𝑥 7 64𝑦 6 128𝑥 7 64𝑦 6 64 × 2 × 𝑥 6 × 𝑥 1 64 × 𝑦 6 8𝑥 3 2𝑥 8𝑦 3 𝑥 3 2𝑥 𝑦 3

Ex. 8 Adding/Subtracting Radicals Must be like radicals Same index and radicand Index – root of radical Radicand – number inside radical Like Unlike 3 ,2 3 3 , 2 5 4 7𝑥 , 4 7𝑥 4 7 , 4 𝑥 or 4 7𝑥 , 5 7𝑥

Ex. 8 cont. 5 7 + 11 −8 7 -3 7 + 11 6 3 𝑥 −2 3 𝑥 4 3 𝑥 10 5 − 20 10 5 − 4 × 5 10 5 −2 5 8 5 12 +6 3 +2 6 4 × 3 +6 3 +2 6 2 3 +6 3 +2 6 8 3 +2 6 Steps 1. Make into like radicals if can Use example 1 notes 2. Combine like radicals Only add or subtract number in front of LEAVE RADICAL ALONE!!

Ex. 9 Multiplying Radicals 5 3 − 75 15 − 375 15 − 25 × 15 15 −5 15 -4 15 3 8 2 +7 32 8 6 +7 96 8 6 +7 16 × 6 8 6 +28 6 When there is a number in front multiply by the square root of the perfect square 36 6 Steps 1. Make sure same index 2. Multiply numbers in front of radical and numbers inside radical Only front with front, inside with inside 3. Make like radicals 4. Combine like radicals

Your Practice 2 5 2 6𝑥 − 96𝑥 4 30𝑥 −2 480𝑥 4 30𝑥 −2 16 × 30𝑥 2 5 2 6𝑥 − 96𝑥 4 30𝑥 −2 480𝑥 4 30𝑥 −2 16 × 30𝑥 4 30𝑥 −8 30𝑥 −4 30𝑥