Math 025 Section 10.3 Radicals

Properties of multiplication and division of radicals · Öb = Öab Öa · Öa = Öa2 = a Öa Ö a b = Öb

Simplify the following: Ö2x2 Ö32x5 = Ö64x7 = Ö64x6Öx = 8x3Öx Ö5(Ö2 – Ö3 ) = Ö10 – Ö15 (Ö2 – 3x)(Ö2 + x) = 2 + xÖ2 – 3xÖ2 – 3x2 = 2 – 2xÖ2 – 3x2

Simplify the following: (Ö5 – 3)(Ö5 + 1) = 5 + Ö5 – 3Ö5 – 3 = 2 – 2Ö5 (Öx – 5)2 (Öx – 5)(Öx – 5) = = x – 5Öx – 5Öx + 25 = x – 10Öx + 25

(Ö5 – 3)(Ö5 + 3) = 5 + 3Ö5 – 3Ö5 – 9 = 5 – 9 = -4 In the preceding example ( Ö5 – 3) and ( Ö5 + 3) are conjugates of each other. Whenever we multiply conjugates, (a – b)(a + b) = a2 – b2 so (x – 3)(x + 3) = x2 – 9 (Öx – 2)(Öx + 2) = x – 4 (Ö5 – Ö8)(Ö5 + Ö8) = 5 – 8 = -3

Simplify the following: (Ö8 – 3)(Ö8 + 3) = 8 – 9 = -1 (Öx + 5) (Öx – 5) = x – 25 (Ö12 – Ö17) (Ö12 + Ö17) = 12 – 17 = -5

A radical expression is not considered to be in simplest form if a radical remains in the denominator. The procedure used to remove a radical from the denominator is called rationalizing the denominator. Simplify: 2 2 Ö3 2Ö3 = = · Ö3 Ö3 Ö3 3 4 4 Ö6 4Ö6 2Ö6 = = = · Ö6 Ö6 Ö6 6 3

Ö Ö Ö Simplify: Ö27a 27a = = Ö9 = 3 Ö3a 3a Ö9xy2 9xy2 y2 = = Ö27x 27x = 3 Ö3a 3a Ö Ö Ö9xy2 9xy2 y2 = = Ö27x 27x 3 Öy2 y Ö3 yÖ3 = = = Ö3 Ö3 Ö3 3

When the denominator contains a radical expression with two terms, simplify by multiplying the numerator and denominator by the conjugate of the original denominator. Simplify: Ö2y Ö2y (Öy – 3) Ö2y2 – 3Ö2y = · = Öy + 3 (Öy + 3) (Öy – 3) y – 9 yÖ2 – 3Ö2y = y – 9

When the denominator contains a radical expression with two terms, simplify by multiplying the numerator and denominator by the conjugate of the original denominator. Simplify: Ö2 Ö2 (Ö2 – Ö6) 2 – Ö12 = = Ö2 + Ö6 (Ö2 + Ö6) (Ö2 – Ö6) 2 – 6 2 – 2Ö3 2(1 – Ö3 ) = = – -4 4 1 – Ö3 = – 2