6-1 Radical Functions & Rational Exponents Unit Objectives: Simplify radical and rational exponent expressions Solve radical equations Solve rational exponent equations Today’s Objective: I can simplify radical expressions.

Review of Exponent Properties 𝑏 𝑚 ⋅ 𝑏 𝑛 = 𝑏 𝑚+𝑛 (𝑏 𝑚 ) 𝑛 = 𝑏 𝑚⋅𝑛 (𝑎𝑏) 𝑛 = 𝑎 𝑛 ⋅ 𝑏 𝑛 𝑏 0 = 1 𝑏 𝑚 𝑏 𝑛 = 𝑏 1 𝑏 𝑛 𝑎 𝑏 𝑛 = 𝑎 𝑛 𝑏 𝑛 𝑚−𝑛 𝑏 −𝑛 = Simplify with positive exponents only. 3 𝑥 −2 𝑦 3 𝑥 5 𝑦 7 −1 2 𝑥 5 ⋅3 𝑥 8 (3 𝑥 5 𝑦 −3 ) 2 9 𝑥 10 𝑦 6 6 𝑥 13 𝑥 7 𝑦 4 3

Roots & Radical Expressions Powers Roots Radicals 2 2 = 4 2 is the square root of 4 4 = 2 2 = 2 2 3 = 8 2 is the cube root of 8 3 8 = 3 2 3 = 2 2 4 = 16 2 is the fourth root of 16 4 16 = 4 2 4 = 2 𝑎 𝑛 = 𝑏 a is the nth root of b 𝑛 𝑏 = 𝑛 𝑎 𝑛 = 𝑎 Index: Degree of root Radicand

Simplifying Radicals 𝑛 𝑎 𝑛 = 𝑎, if 𝑛 is odd 𝑎 , if 𝑛 is even 3 𝑥 12 = Write radicand in factors raised to the nth power or less. Take the nth root of all factors to the nth power. Simplify in front of radical and under radical. 3 𝑥 12 = 3 𝑥 3 ⋅ 𝑥 3 ⋅ 𝑥 3 ⋅ 𝑥 3 25 = 5 2 = |5| =5 3 125 = 3 5 3 = =𝑥⋅𝑥⋅𝑥⋅𝑥 = 𝑥 4 5 3 −27 = 3 (−3) 3 = −3 5 32 𝑥 10 = 5 2 5 ⋅ 𝑥 5 ⋅ 𝑥 5 3 64 = 3 8⋅8 =2⋅𝑥⋅𝑥 = 2𝑥 2 = 3 2 3 ⋅ 2 3 = 2⋅2= 4

Simplifying Radicals 𝑛 𝑎 𝑛 = 𝑎, if 𝑛 is odd 𝑎 , if 𝑛 is even Write radicand in factors raised to the nth power or less. Take the nth root of all factors to the nth power. Simplify in front of radical and under radical. 3 54 𝑥 5 = 3 27⋅2⋅ 𝑥 3 ⋅ 𝑥 2 24 = 4⋅6 2 2 ⋅6 = 2 6 = 3 3 3 ⋅2⋅ 𝑥 3 ⋅ 𝑥 2 =3𝑥 3 2 𝑥 2 3 24 = 3 8⋅3 4 48 𝑥 13 = 4 16⋅3⋅ 𝑥 4 ⋅ 𝑥 4 ⋅ 𝑥 4 ⋅𝑥 3 2 3 ⋅3 = 2 3 3 = 4 2 4 ⋅3⋅ 𝑥 4 ⋅ 𝑥 4 ⋅ 𝑥 4 ⋅𝑥 Homework: worksheet =2⋅|𝑥|⋅|𝑥|⋅|𝑥|⋅ 4 3𝑥 =2 |𝑥 3 | 4 3𝑥