Math 002 Unit 5  Student Notes


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1 Sections 7.1 Radicals and Radical Functions Math 002 Unit 5  Student Notes Objectives: Find square roots, cube roots, nth roots. Find where a is a real number. Look at the graphs of square root and cube root functions and evaluate function values. 1. Finding Square Roots: EX 1 Radical expression: To take a square root means to undo a square. If a is a nonnegative number, then o is the principal, or nonnegative, square root of a. o is the negative square root of a. Example 1: Simplify. Assume that all variables represent positive numbers. a) b) c) d) e) f) 2. Finding Roots. EX 3, EX 4 Radical expression: To take an root means to undo an nth power. 1
2 Example 2: Simplify each radical. Assume that all variables represent positive real numbers. The index is important. #59 #60 #64 #65 #71 Note: is a rational number. is an irrational number. 3. The square root and cube root functions. EX 6, EX 7, EX 8 Domain. General Graph: Basic square root function Basic cube root function Example 3: Match the graph with its equation. Also, give the domain. A. B. C. D. Example 4: If and, find each function value. a. b. c. d. 2
3 Section Rational Exponents Objectives: Understand the meaning of, and. Use rules for exponents to simplify expressions that contain rational exponents. Use rational exponents to simplify radical expressions. Sometimes we will see radicals expressed as rational exponents. 1. Understand the meaning of. EX 1 If a is a real number and n is a positive integer, then. The quantity is the nth root of a. Notice that the denominator of the rational exponent corresponds to the of the radical. Example 1: Use radical notation to write each expression. Simplify if possible. a. b. c. d. e. f. 2. Understand the meaning of. EX 2 By properties of exponents, We can generalize this: If m and n are positive integers greater than 1 (with in lowest terms), then Example 2: Use radical notation to write each expression. Simplify if possible. a. b. c. d. 3
4 3. Understand the meaning of. EX 3 Ask students, What is? What is? What is when n is an integer? What is where m and n are integers? Example 3: Write each expression with a positive exponent, and then simplify. a. b. c. d. 4. Using rules for exponents to simplify expressions. EX 4, EX 5 The properties of exponents which we reviewed earlier, apply to rational exponents as well. Let a and b be real numbers Let m and n be integers Let m and n be rational numbers Example 4: Use the properties of exponents to simplify each expression. Write with positive exponents. a. b. c. d. e. #60. Example 5: Use rational exponents to simplify. Assume that variables represent positive numbers. #76. #80. #83. 4
5 Section 7.3 Simplifying Radical Expressions Objectives: Use the product rule for radicals. Use the quotient rule for radicals. Simplify radicals. Use the distance formula. 1. Using the Product Rule. EX 1 Product Rule for Radicals If two rational expressions have the same then when multiplying radicals we can simply multiply the values under the radicals. Example 1: a. b. c. d. 2. Using the Quotient Rule. EX 2 Quotient Rule for Radicals Example 2: a. b. c. d. 3. Simplifying Radicals EX 3, EX 4, EX 5 Both the product and quotient rules can be used to simplify a radical. Example 3: Simplify. Assume variables represent nonnegative values. a. b. c. d. e. f. 5
6 g. h. A radicand in the form is simplified when the radicand a contains no factors that are perfect nth powers (other than 1 or 1). All factors of the radicand have exponents less than the index. The greatest common factor of the index and exponents of all the radicand factors is The distance formula. EX 6 Now that we know how to simplify radicals, we can derive and use the distance formula. The distance between two points and is given by: Example 4: Find the distance between each pair of points. Give an exact distance and a threedecimal approximation. a. #75. 6
7 Section 7.4 Adding, Subtracting, and Multiplying Radical Expressions Objectives: Add/subtract radical expressions. Multiply radical expressions. 1. Like radicals: radicals with the same and the same. Complete the table with like or unlike Terms Like or unlike terms? 2. Add/Subtract Radical Expressions. EX 1, EX 2, EX 3 Simplify each radical term first Combine like terms (like radicals). Example 2: Add or subtract as indicated. a. b. c. d. e. f. 7
8 #15 # Multiplying Radical Expressions. EX 4 As with multiplying polynomials, many of the same properties are used to multiply expressions with radical terms. Example 3: Multiply. a. b. c. #56. d. e. 8
9 Section 7.5 Rationalizing Denominators of Radical Expressions Objectives: Rationalize denominators with one term. Rationalize denominators having two terms. Rationalizing the Denominator means rewriting the rational expression without any in the denominators. 1. Rationalizing the denominator when there is one term in the denominator. EX 1, EX 2, EX 3 Multiply the numerator and denominator by the expression that will make the denominator the nth root of a perfect nth power. Simplify. Example 1: Rationalize the denominator. a. b. c. d. e. f. Quick Review: These are called conjugates. Because the product results in the difference of two squares, we will need the conjugates to rationalize the denominator when there are two terms in the denominators. Example 2: Find the conjugate of each expression, then find their product. expression conjugate product
10 2. Rationalizing Denominators Having Two Terms. EX 4 Multiply the numerator and denominator by the conjugate of the denominator. Simplify. Example 3: Rationalize each denominator. a. b. c. #48. 10
11 Section 7.6 Radical Equations and Problem Solving Objectives: Solve equations that contain radical expressions. Use the Pythagorean theorem to model problems. 1. POWER RULE EX 1, EX 4, EX 5 If both sides of an equation are raised to the same power, all solutions of the original equation are among solutions of the new equation. This property does not state that raising both sides of an equation to a power yields an equivalent equation. A solution of the new equation may or may not be a solution of the original equation. You must check each solution of the new equation to make sure it is a true solution or extraneous solution of the original equation. Example 1: Solve. Show checking both numerically and graphically. a) Check: b) c) d) e) 11
12 # Pythagorean Theorem. EX 6 If and are the lengths of the legs of a right triangle and is the length of the hypotenuse, then. #53. Figure is of a right triangle with side length 3m and hypotenuse 7m find the length of the other leg. 12
13 Section 7.7 Complex Numbers Objectives: Write square roots of negative numbers in the formbi. Add, subtract, multiply, and divide complex numbers. Raise I to powers. 1. Write Square Roots of Negative numbers in the form. EX 1 The imaginary unit, written as, is the whole number whose square is 1. and. Example 1: Write in terms of Encourage students to write the radical last when simplifying. a) b) c. Multiplying or dividing with a negative number under a square root. EX 2 Example 2: Multiply or divide. Emphasize that the following is not true:. d) e) f) g) h) 2. Complex Numbers. A complex number is a number that can be written in the form, where are real numbers and is the imaginary unit. Example 3: Write the following in complex form. a. b. c. d. 13
14 Addition and subtraction of complex numbers Add/Subtract like terms Example 4: Perform the indicated operations. a. b. Products of complex numbers Multiply as polynomials, rewrite to basic terms and then combine like terms. c) d) e) Quotients of complex numbers Multiply numerator and denominator by the conjugate of the denominator; rewrite in complex form; simplify f) g) h) POWERS of 14
15 Section 8.1 Solving Quadratic Equations by Completing the Square Objectives: Use the square root property to solve quadratic equations. Use quadratic equations to solve problems. We will continue to look at different ways to solve quadratic equations. Previously, we learned how to solve a quadratic equation by factoring. In the next two sections we will look at different ways to solve quadratic equations 1. The Square Root Property. EX 1, EX 2, EX 3, EX 4 If is a real number and if, then. Even when the right side of the equation is not a perfect square, we can use this property. Example 1: Solve for x. a. b. c. d. Review: Perfect Square Trinomials Factored Form 2. Completing the Square. EX 5, EX 6, EX 7 This method is such a useful tool to solve quadratic equations, we can change equations that are not perfect squares on the lefthand sides to perfect squares. Example 2: What number is needed to make a perfect square trinomial? a. b. c. d. 15
16 Example 3: Solve by completing the square. Leading coefficient is 1. i. j. Leading coefficient is not 1. k. l. Example 4: Use the graph to determine how many real number solution(s) exists for each equation. #72. #74. 16
17 Section 8.2 Solving Quadratic Equations by the Quadratic Formula Objectives: Solve quadratic equations by using the quadratic formula. Determine the number and type of solutions of a quadratic equation by using the discriminant. Solve geometric problems modeled by quadratic equations. Any quadratic equation can be solved by completing the square. When we complete the square of the general quadratic equation in standard form we obtain the Quadratic Formula. 1. Quadratic Formula. EX 1, EX 2 A quadratic equation written in the form has the solutions Example 1: Solve. a. b. c. What are the different possibilities of solutions for a quadratic equation? 17
18 2. The Discriminant. EX 5 The discriminant is the portion of the quadratic formula under the radical:. It can tell us what the solutions are going to look like before we solve them. Positive Zero Negative Number and Type of solutions Example 2: Use the discriminant to determine the number and types of solutions of each equation. You can check with the graphing calculator. d. e. f. 3. Applications. #52. Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk. 54. The hypotenuse of an isosceles right triangle is one meter longer than either of its legs. Find the length of each side. 18
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