10-1 Simplifying Radicals Hubarth Algebra

Radical Expressions like 2 3 and 𝑥+3 contain a radical. Multiplication Property of Square Roots For every number 𝑎≥0 𝑎𝑛𝑑 𝑏≤0, 𝑎𝑏 = 𝑎 ∙ 𝑏 . EXAMPLE 54 = 9 ∙ 6 =3∙ 6 =3 6

Ex 1 Simplifying Square Roots a. 45 b. 48 c. 243 9 ∙ 5 16 ∙ 3 81 ∙ 3 3 ∙ 5 3 5 4∙ 3 4 3 9∙ 3 9 3

Ex 2 Remove Variable Factors Simplify 𝑎. 27 𝑥 5 b. 28 𝑥 7 c. 16 𝑎 12 9 𝑥 4 ∙ 3𝑥 4 𝑥 6 ∙ 7𝑥 4 𝑎 6 3𝑥 2 3𝑥 2𝑥 3 7𝑥

Ex 3 Multiplying Two Radicals Simplify each radical expression. a. 12 • 32 b. 7 5x • 3 8x 12∙32 21 40 𝑥 2 384 21 4 𝑥 2 ∙ 10 64 ∙ 6 21∙2𝑥∙ 10 8 6 42𝑥 10

Division Property of Square Roots For every number 𝑎≥0 𝑎𝑛𝑑 𝑏>0, 𝑎 𝑏 = 𝑎 𝑏 EXAMPLE 16 25 = 16 25 = 4 5

Ex 4 Simplifying Fractions Within Radicals Simplify each radical expression. a. 13 64 b. 49 x4 = 13 64 = 13 8 = 49 x4 7 x2 =

Ex 5 Simplifying Radicals by Dividing Simplify each radical expression. a. 120 10 b. 75x5 48x 120 10 = 12 = 75x5 48x 25x4 16 = 4 • 3 = 25x4 16 = 4 • 3 = 25 • x4 16 = 2 3 5x2 4 =

Ex 6 Rationalizing a Denominator Simplify by rationalizing the denominator. a. 3 7 b. 11 12x3 3 7 √ 7 = • √ 3x 11 12x3 = • 3 7 49 = 33x 36x4 = 3 7 7 = 33x 6x2 =

Practice 1. Simplify each radical expression. a. 50 b. 54 𝑥 9 𝑦 10 c. 2 5𝑎 2 ∙6 10 𝑎 3 d. 144 9 e. 25 𝑝 3 𝑞 2 2. Rationalize the denominator. a. 3 3 b. 4 3 2 5 2 3 𝑥 4 𝑦 5 6𝑥 60 𝑎 2 2𝑎 4 5𝑝 𝑝 𝑞 3 2 6