 # 7.1, 7.2 & 7.3 Roots and Radicals and Rational Exponents Square Roots, Cube Roots & Nth Roots Converting Roots/Radicals to Rational Exponents Properties.

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7.1, 7.2 & 7.3 Roots and Radicals and Rational Exponents Square Roots, Cube Roots & Nth Roots Converting Roots/Radicals to Rational Exponents Properties of Exponents Apply to Rational Exponents Too! Simplifying Radical Expressions  Multiplying  Dividing Try graphing : y =  3 x-1 And  x y =

Square Roots & Cube Roots A number b is a square root of a number a if b 2 = a  25 = 5 since 5 2 = 25 Notice that 25 breaks down into 5 5 So,  25 =  5 5 See a ‘group of 2’ -> bring it outside the radical (square root sign). Example:  200 =  2 100 =  2 10 10 = 10  2 A number b is a cube root of a number a if b 3 = a  8 = 2 since 2 3 = 8 Notice that 8 breaks down into 2 2 2 So,  8 =  2 2 2 See a ‘group of 3’ –> bring it outside the radical (the cube root sign) Example:  200 =  2 100 =  2 10 10 =  2 5 2 5 2 =  2 2 2 5 5 = 2  25 3 3 3 3 3 3 3 3 Note:  -25 is not a real number since no number multiplied by itself will be negative Note:  -8 IS a real number (-2) since -2 -2 -2 = -8 3

Nth Root ‘Sign’ Examples  16  -16 = 4 or -4 not a real number  -16 4 not a real number Even radicals of negative numbers Are not real numbers.  -32 5 = -2 Odd radicals of negative numbers Have 1 negative root.  32 5 = 2 Odd radicals of positive numbers Have 1 positive root. Even radicals of positive numbers Have 2 roots. The principal root Is positive.

Exponent Rules (XY) m = x m y m XYXY m = XmYmXmYm

Examples to Work through

Product Rule and Quotient Rule Example

Some Rules for Simplifying Radical Expressions

Example Set 1

Example Set 2

Example Set 3

7.4 & 7.5: Operations on Radical Expressions Addition and Subtraction (Combining LIKE Terms) Multiplication and Division Rationalizing the Denominator

Rationalizing the Denominator Remove all radicals from the denominator

Rationalizing Continued… Multiply by the conjugate

7.6 Solving Radical Equations X 2 = 64 #1 #2 #3 #4

Radical Equations Continued… Example 1: x +  26 – 11x = 4  26 – 11x = 4 - x (  26 – 11x) 2 = (4 – x) 2 26 – 11x = (4-x) (4-x) 26 - 11x = 16 –4x –4x +x 2 26 –11x = 16 –8x + x 2 -26 +11x 0 = x 2 + 3x -10 0 = (x - 2) (x + 5) x – 2 = 0 or x + 5 = 0 x = 2 x = -5 Example 2:  3x + 1 –  x + 4 = 1  3x + 1 =  x + 4 + 1 (  3x + 1) 2 = (  x + 4 + 1) 2 3x + 1 = (  x + 4 + 1) (  x + 4 + 1) 3x + 1 = x + 4 +  x + 4 +  x + 4 + 1 3x + 1 = x + 4 + 2  x + 4 + 1 3x + 1 = x + 5 + 2  x + 4 -x -5 -x -5 2x - 4 = 2  x + 4 (2x - 4) 2 = (2  x + 4) 2 4x 2 –16x +16 = 4(x+4) 4x 2 –20x = 0 4x(x –5) = 0, so…4x = 0 or x – 5 = 0 x = 0 or x = 5 4x+16

7.7 Complex Numbers REAL NUMBERS Imaginary Numbers Irrational Numbers ,  8, -  13 Rational Numbers (1/2 –7/11, 7/9,.33 Integers (-2, -1, 0, 1, 2, 3...) Whole Numbers (0,1,2,3,4...) Natural Numbers (1,2,3,4...)

Complex Numbers (a + bi) Real Numbers a + bi with b = 0 Imaginary Numbers a + bi with b  0 i =  -1 where i 2 = -1 Irrational Numbers Rational Numbers Integers Whole Numbers Natural Numbers

Simplifying Complex Numbers A complex number is simplified if it is in standard form: a + bi Addition & Subtraction) Ex1: (5 – 11i) + (7 + 4i) = 12 – 7i Ex2: (-5 + 7i) – (-11 – 6i) = -5 + 7i +11 + 6i = 6 + 13i Multiplication) Ex3: 4i(3 – 5i) = 12i –20i 2 = 12i –20(-1) = 12i +20 = 20 + 12i Ex4: (7 – 3i) (-2 – 5i) [Use FOIL] -14 –35i +6i +15i 2 -14 –29i +15(-1) -14 –29i –15 -29 –29i

Complex Conjugates The complex conjugate of (a + bi) is (a – bi) The complex conjugate of (a – bi) is (a + bi) (a + bi) (a – bi) = a 2 + b 2 Division 7 + 4i 2 – 5i 2 + 5i 14 + 35i + 8i + 20i 2 14 + 43i +20(-1) 2 + 5i 4 + 10i –10i – 25i 2 4 –25(-1) 14 + 43i –20 -6 + 43i -6 43 4 + 25 29 29 29 == = + i=

Square Root of a Negative Number  25  4 =  100 = 10  -25  -4 =  (-1)(25)  (-1)(4) =  (i 2 )(25)  (i 2 )(4) = i  25 i  4 = (5i) (2i) = 10i 2 = 10(-1) = -10 Optional Step

Practice – Square Root of Negatives

Practice – Simplify Imaginary Numbers i 2 = i 3 = i 4 = i 5 = i 6 = -i 1 i i 0 = 1 i 1 = i Another way to calculate i n Divide n by 4. If the remainder is r then i n = i r Example: i 11 = __________ 11/4 = 2 remainder 3 So, i 11 = i 3 = -i

Practice – Simplify More Imaginary Numbers

Practice – Addition/Subtraction 10 +8i -4 +10i

Practice – Complex Conjugates Find complex conjugate. 3i => -4i =>

Practice Division w/Complex Conjugates 4__ 2i =

Things to Know for Test 1.Square Root, Cube Root, Nth Root - Simplify 2.Rational Exponents – Convert back and forth to/from radical form 3.Add, Subtract, Multiply & Divide radicals & rational exponents 4.Rationalize denominator 5.Solve radical equations 6.Imaginary Numbers – Add, subtract, multiply, divide 7.Imaginary Numbers – find the value of i n

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