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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 =

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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

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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.

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

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Examples to Work through

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Product Rule and Quotient Rule Example

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Some Rules for Simplifying Radical Expressions

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Example Set 1

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Example Set 2

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Example Set 3

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

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Radical Operations with Numbers

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Radical Operations with Variables

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Multiplying Radicals (FOIL works with Radicals Too!)

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Rationalizing the Denominator Remove all radicals from the denominator

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Rationalizing Continued… Multiply by the conjugate

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

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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

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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...)

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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

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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

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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=

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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

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Practice – Square Root of Negatives

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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

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Practice – Simplify More Imaginary Numbers

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

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

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Practice Division w/Complex Conjugates 4__ 2i =

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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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