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Unit 4

Unit 4. Richardson. Bellringer 9/23/14. Simplify : Simplify :. Simplifying Radicals Review and Radicals as Exponents. A radical expression contains a root, which can be shown using the radical symbol, .

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Unit 4

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  1. Unit 4 Richardson

  2. Bellringer9/23/14 • Simplify: • Simplify:

  3. Simplifying Radicals Review and Radicals as Exponents

  4. A radical expression contains a root, which can be shown using the radical symbol, . • The root of a number x is a number that, when multiplied by itself a given number of times, equals x. • For Example: Simplifying Radicals Basic Review

  5. Simplifying Radicals Steps • Use a factor tree to put the number in terms of its prime factors. • Group the same factor in groups of the number on the outside. • Merge those numbers into 1 and place on the outside. • Multiply the numbers outside together and the ones left on the inside together. 6

  6. To add and/or subtract radicals you must first Simplify them, then combine like radicals. • Ex: Simplifying Radicals Adding and Subtracting

  7. Square Roots as Exponents Square Root Exponent Please put this in your calculator. What did you get? = 9 3*3

  8. Bellringer 9/24/14 Please get the calculator that has your seat number on it, if there isn’t one please see me! • Simplify: • Rewrite as an exponent and solve on your calculator:

  9. Exponent Rules and Imaginary Numbers - with multiplying and dividing square roots if we have time

  10. Imaginary Numbers • Can you take the square root of a negative number? • Ex: → what number times itself () gives you a negative 4? • Can u take the cubed root of a negative number? • Ex: → what number times itself, and times () itself again gives you a negative 8? • The imaginary unit iis used to represent the non-real value, . • An imaginary number is any number of the form bi, where b is a real number, i= , and b ≠ 0.

  11. Exponent Rules Zero Exponent Property Negative Exponent Property A negative exponent of a number is equal to the reciprocal of the positive exponent of the number. • A base raised to the power of 0 is equal to 1. • a0 = 1

  12. Exponent Rules Product of Powers Property Quotient of Powers Property To divide powers with the same base, subtract the exponents. • To multiply powers with the same base, add the exponents.

  13. Exponent Rules Power of a Power Property Power of a Product Property To find the power of a product, distribute the exponent. • To raise one power to another power, multiply the exponents.

  14. Exponent Rules Power of a Quotient Property • To find the power of a quotient, distribute the exponent.

  15. Bellringer 9/25/14 • Simplify: • Simplify:

  16. Imaginary Numbers and Exponents 1 And so on…

  17. Roots and Radicals Review The Rules (Properties) Multiplication Division b may not be equal to 0.

  18. Roots and Radicals The Rules (Properties) Multiplication Division b may not be equal to 0.

  19. Roots and Radicals Review Examples: Multiplication Division

  20. Roots and Radicals Review Examples: Multiplication Division

  21. Intermediate Algebra MTH04 Roots and Radicals To add or subtract square roots or cube roots... • simplify each radical • add or subtract LIKE radicals by • adding their coefficients. Two radicals are LIKE if they have the same expression under the radical symbol.

  22. Complex Numbers

  23. Complex Numbers • All complex numbers are of the form a + bi, where a and b are real numbers and iis the imaginary unit. The number a is the real part and bi is the imaginary part. • Expressions containing imaginary numbers can also be simplified. • It is customary to put I in front of a radical if it is part of the solution.

  24. Simplifying with Complex Numbers Practice • Problem 1 • Problem 2

  25. Bellringer 9/26/14 • Sub Rules Apply

  26. Practice With Sub – simplify, i, complex, exponent rules

  27. Bellringer 9/29/14 • Write all of these questions and your response • Is this your classroom? • Should you respect other people’s property and work space? • Should you alter Mrs. Richardson’s Calendar? • How should you treat the class set of calculators?

  28. Review Practice Answers Discuss what to do when there is a substitute

  29. Bellringer9/30/14*EQ- What are complex numbers? How can I distinguish between the real and imaginary parts? • 1. How often should we staple our papers together? • When should we turn in homework and where? • When and where should we turn in late work? 4. What are real numbers?

  30. Let’s Review the real number system! • Rational numbers • Integers • Whole Numbers • Natural Numbers • Irrational Numbers

  31. More Examples of The Real Number System

  32. Now we have a new number! Complex Numbers Defined. • Complex numbers are usually written in the form a+bi, where a and b are real numbers and iis defined as . Because does not exist in the set of real numbers I is referred to as the imaginary unit. • If the real part, a, is zero, then the complex number a +bi is just bi, so it is imaginary. • 0 + bi = bi , so it is imaginary • If the real part, b, is zero then the complex number a+bi is just a, so it is real. • a+ 0i =a , so it is real

  33. Examples • Name the real part of the complex number 9 + 16i? • What is the imaginary part of the complex numbers 23 - 6i?

  34. Check for understanding • Name the real part of the complex number 12+ 5i? • What is the imaginary part of the complex numbers 51 - 2i? • Name the real part of the complex number 16i? • What is the imaginary part of the complex numbers 23?

  35. Name the real part and the imaginary part of each. 1. 2. 3. 4. 5.

  36. Bellringer 10/1/14*EQ- How can I simplify the square root of a negative number? For Questions 1 & 2, Name the real part and the imaginary part of each. 1. 2. For Questions 3 & 4, Simplify each of the following square roots. 3. 4.

  37. Simply the following Square Roots.. 1. 2. 3. 4. How would you take the square root of a negative number??

  38. Simplifying the square roots with negative numbers • The square root of a negative number is an imaginary number. • You know that i = • When n is some natural number (1,2,3,…), then

  39. Simply the following Negative Square Roots.. 1. 2. 3.

  40. Let’s review the properties of exponents….

  41. How could we make a list of ivalues?

  42. Practice • Simply the following Negative Square Roots.. 1. 2. 3. • Find the following i values.. 4. 5.

  43. Bellringer 10/2/14 Simply the following Negative Square Roots: 1. 2. 3.

  44. How could we make a list of ivalues?

  45. Note: • A negative number raised to an even power will always be positive • A negative number raised to an odd power will always be negative.

  46. How could we make a list of ivalues?

  47. Bellringer 10/3/14 • Turn in your Bellringers

  48. Bellringer 10/13/14 • Simplify the following:

  49. Review

  50. Review – Work on your own paper

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