1. Math Section Radicals

2. Properties of multiplication and division of radicalsÖb =
Öab Öa Öa =
Öa2 = a Öa Ö
a b = Öb

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

4. 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

5. (Ö5 – 3)(Ö5 + 3)
= Ö5 – 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

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

7. 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

8. 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

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:
Ö2y
Ö2y
(Öy – 3)
Ö2y2 – 3Ö2y = =
Öy + 3
(Öy + 3)
(Öy – 3)
y – 9
yÖ2 – 3Ö2y =
y – 9

10. 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