# Simplifying Radicals - OGLESBY MATH Warm-up 11 August 2017 List all of the positive factors (factor tree). 1. 400 2. 85 3. 333 Simplifying Radicals Perfect Squares 1 4 9 16 25 36 49 64 81 100

121 144 169 196 225 256 289 324 361 400 625 4 =2 16 =4

25 =5 100 = 10 144 = 12 LEAVE IN RADICAL FORM Perfect Square Factor * Other Factor 8 = 4*2 =

2 20 = 4*5 = 2 5 32 = 16 * 2 = 4

75 = 25 * 3 = 5 3 40 = 4 *10 = 2 10 2 2 LEAVE IN RADICAL FORM Perfect Square Factor * Other Factor

48 = 16 * 3 = 4 3 80 = 16 * 5 = 4 5 50 =

25 * 2 = 25 2 125 = 25 * 5 = 450 = 225 * 2 = 15 2 5 5 + To combine radicals: combine the coefficients of like radicals Simplify each expression

6 7 5 7 3 7 8 7 5 6 3 7 4 7 2 6 3 6 7 7 Simplify each expression: Simplify each radical first and then combine. 2 50 3 32 2 25 * 2 3 16 * 2 2 *5 2 3* 4 2 10 2 12 2 2 2 Simplify each expression: Simplify each radical first and then combine. 3 27 5 48 3 9 * 3 5 16 * 3

3*3 3 5* 4 3 9 2 20 2 29 2 LEAVE IN RADICAL FORM Perfect Square Factor * Other Factor 18 = = 288 = = 75

= = 24 = = 72 = = Simplify each expression 6 5 5 6 3 6 3 24 7 54

2 8 7 32 Simplify each expression 6 5 5 20 18 7 32 2 28 7 6 63 * To multiply radicals: multiply the coefficients and then multiply the radicands and then simplify the remaining radicals. Multiply and then simplify 5 * 35 175 25 * 7 5 7 2 8 * 3 7 6 56 6 4 *14 6 * 2 14 12 14

2 5 * 4 20 20 100 20 *10 200 5 2 5* 5 25 5 7* 7 49 7 8* 8 64 8 x x* x

x x 7 2 8 2 2 2 To divide radicals: divide the coefficients, divide the radicands if possible, and

rationalize the denominator so that no radical remains in the denominator 56 8 7 4*2 2 2 This cannot be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something

so we can eliminate the radical in the denominator. 6 7 6 * 7 42 49 7 7 42 7

42 cannot be simplified, so we are finished. This can be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something so we can eliminate the radical in the denominator. 5 10

1 * 2 2 10 2 2 This cannot be divided which leaves the radical in the denominator. We do not leave radicals in the denominator. So we need to rationalize by multiplying the fraction by something

so we can eliminate the radical in the denominator. 3 12 3 * 12 3 3 3 3 36 Reduce

the fraction. 3 3 6 3 6 X Y 4 =X 2 = Y3

6 6 2 P X Y 4 4X Y 8 2 25C D 10 = P2X3Y = 2X2Y = 5C4D10

X 3 = X = Y 5 2 X *X

X = Y = 2 Y 4 Y Y 3 PX Y

= 3 2 X Y = XY 7 12 X Y 8 2 9 =

25C D = Y Y 5 5 2 * PXY PXY