1/8/2016Math KM1 Chapter 8: Radical Expressions, Equations, and Functions 8.1 Radical Expressions & Functions 8.2 Rational Numbers as Exponents 8.3 Simplifying Radical Expressions 8.4 Addition, Subtraction, and more Multiplication 8.5 More on Division of Radical Expressions 8.6 Solving Radical Equations 8.7 Applications involving Powers and Roots 8.8 The Complex Numbers

1/8/2016Math KM2 8.1 Radical Expressions & Functions

1/8/2016Math KM3 Index The “default” index is 2. Radicand Parts of a Radical means 8.1

1/8/2016Math KM4 What’s the difference? What are the square root(s) of 49? The square root(s) of 49 are -7 and 7. Every number (except zero) has two square roots. 8.1

1/8/2016Math KM5 Is the radical different? Well then, what is ? is the principal square root of

1/8/2016Math KM6 Counting with Radicals 8.1

1/8/2016Math KM7 Simplify a few? Not a Real Number 8.1

1/8/2016Math KM8 Exact vs Approximate is an EXACT value. is an APPROXIMATE value. 8.1

1/8/2016Math KM9 The Square Root Function The DOMAIN is x > 0 xf(x)

1/8/2016Math KM10 A minor change? The DOMAIN is x > 2 xf(x)

1/8/2016Math KM11 Random Radicals ? 8.1

1/8/2016Math KM12 Tricky Problems? Absolutely! 6.1

1/8/2016Math KM Rational Numbers as Exponents 8.2

1/8/2016Math KM14 The Basic Idea! 8.2

1/8/2016Math KM15 See How this Works? 8.2

1/8/2016Math KM16 Let’s try another! 8.2

1/8/2016Math KM17 Isn’t this fun? 8.2

1/8/2016Math KM18 Negative? OK! 6.2

1/8/2016Math KM19 Try this one? 8.2

1/8/2016Math KM20 Where We Left Off Last Class

1/8/2016Math KM21 Simplify? 8.2

1/8/2016Math KM22 Play by the Rules! 8.2

1/8/2016Math KM23 What if? 8.2

1/8/2016Math KM24 You can do this! 8.2

1/8/2016Math KM25 Rewrite in radical form Reverse the Process? 8.2

1/8/2016Math KM26 Rewrite in exponential form OK...now the other way! 8.2

1/8/2016Math KM27 Conquer This! 8.2

1/8/2016Math KM Simplifying Radical Expressions 8.3

1/8/2016Math KM29 Simplify: Assume that all expressions under radicals represent nonnegative numbers

1/8/2016Math KM30 Simplify:

1/8/2016Math KM31 Simplify:

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1/8/2016Math KM34 Product Property are Real numbers

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1/8/2016Math KM36 8.3

1/8/2016Math KM37 Quotient Property are Real numbers 8.3

1/8/2016Math KM38 8.3

1/8/2016Math KM Addition, Subtraction, and more Multiplication 8.4

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1/8/2016Math KM42 8.4

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1/8/2016Math KM44 8.4

1/8/2016Math KM45 8.4

1/8/2016Math KM46 8.4

1/8/2016Math KM More on Division of Radical Expressions 8.5

1/8/2016Math KM48 8.5

1/8/2016Math KM49 8.5

1/8/2016Math KM50 8.5

1/8/2016Math KM51 8.5

1/8/2016Math KM52 8.5

1/8/2016Math KM Solving Radical Equations 8.6

1/8/2016Math KM54 ISOLATE the RADICAL Raise to the power of the index Check for extraneous solutions More than one radical?… separate the radicals to opposite sides of the equation and power up!! 8.6

1/8/2016Math KM55 An equation that contains a variable expression in a radicand is a radical equation. A POWER RULE for EQUATIONS 8.6

1/8/2016Math KM56 8.6

1/8/2016Math KM57 8.6

1/8/2016Math KM58 8.6

1/8/2016Math KM59 8.6

1/8/2016Math KM60 8.6

1/8/2016Math KM61 8.6

1/8/2016Math KM62 Be sure to check your solutions! 8.6

1/8/2016Math KM Applications involving Powers and Roots 8.7

1/8/2016Math KM64 An object is dropped from a bridge. Find the distance the object has fallen when its speed reaches 120 ft/s. Use the equation, where v is the speed of the object in feet per second an d is the distance in feet. 8.7

1/8/2016Math KM65 An 18 foot ladder is leaning against a building. How high on the building will the ladder reach when the base of the ladder is 6 feet from the building? 8.7

1/8/2016Math KM The Complex Numbers 8.8

1/8/2016Math KM67 The original i... The term “imaginary number” was coined in 1637 by Rene Descartes Several subjects in physics require complex numbers, such as quantum mechanics, general relativity and fluid dynamics. Also, complex numbers play a key role in chaos theory and in fractal geometry. Imaginary numbers were defined in 1572 by Rafael Bombelli an Italian mathematician.Rafael Bombelli 8.8

1/8/2016Math KM68 More uses for i... Complex numbers are used extensively in physics to describe Electromagnetic Waves and Quantum Mechanics. In electrical engineering complex numbers are used to represent the phase of an alternating signal affected by inductance and capacitance. However, the actual voltage or current at any time is still a real number (which is calculated from the complex number)

1/8/2016Math KM69 Aerodynamics too… The mapping function gives the velocity and pressures around the airfoil. Knowing the pressure around the airfoil, allows the “lift” to be determined

1/8/2016Math KM70 Imaginary... Not really! “Mathematically, the Mandelbrot set can be defined as the set of complex c-values for which the orbit of 0 under iteration of the complex quadratic polynomial x n+1 =x n 2 + c remains bounded.”

1/8/2016Math KM71 o The Complex Plane / Unit Circle Imaginary axis Real axis 8.8

1/8/2016Math KM72 Imaginary axis Real axis Powers of “i” ? Use the i-clock 8.8

1/8/2016Math KM73 Use the i-clock 8.8

1/8/2016Math KM74 Use your i-magination! 8.8

1/8/2016Math KM75 Try this one?

1/8/2016Math KM76 Complex it is! 8.8

1/8/2016Math KM77 But not too complex for you!

1/8/2016Math KM78 ( i ) + ( 9 + 3i ) i bet you can do this! = i 8.8

1/8/2016Math KM79 ( i ) – ( 7 – 3i ) No problem...right? = i 8.8

1/8/2016Math KM80 ( 3 + 2i ) + ( 3 – 2i ) Add Complex Conjugates? Really? = 6 These numbers add to 6 and multiply to

1/8/2016Math KM81 Rewrite, then Simplify

1/8/2016Math KM82 (3i)(5i) Multiply? i remember! = 15i 2 =

1/8/2016Math KM83 (-8i)(7i) Here’s another! = -56i 2 =

1/8/2016Math KM84 (-6i)(-2i) One More? = 12i 2 =

1/8/2016Math KM85 ( 4 + 3i)(5 – i) FOiL... i know you can! = 20 – 4i + 15i – 3i 2 = i = 20 – 4i + 15i

1/8/2016Math KM86 (4 + 3i)(4 – 3i) Product of Complex Conjugates = 16 – 9i 2 = = 25 These numbers multiply to 25 and add to 8 8.8

1/8/2016Math KM87 Convert to i then Distribute 8.8

1/8/2016Math KM88 “Real-ize” the denominator! 8.8

1/8/2016Math KM89 Another Denominator to “Realize”. 8.8

1/8/2016Math KM90 Conjugate Time! 8.8

1/8/2016Math KM91 Another Reality Check? 8.8

1/8/2016Math KM92 Solution Check? Is 1 + 2i a solution of x 2 – 2x + 5 = 0 ? 8.8

1/8/2016Math KM93 Seattle Fractals Amazing Seattle Fractals! Fractal Art, Screensavers, Tutorials, Software & more! Doug Harrington

1/8/2016Math KM94 That’s All For Now!