Slide 1 / 180 Radicals and Rational Exponents Slide 2 / 180 Roots and Radicals Table of Contents: Square Roots Intro to Cube Roots n th Roots Irrational Roots Rational Exponents Operations with Radicals Addition and Subtraction Multiplication Division: Rationalizing the Denominator Complex Numbers Slide 3 / 180 Roots
Slide 4 / 180 The symbol for taking a square root is , it is a radical sign. The square root cancels out the square. There is no real square root of a negative number. is not real (4 2 =16 and (-4) 2 =16) Slide 5 / 180 1 What is 1 ? Slide 6 / 180 2 What is ?
Slide 7 / 180 3 What is ? Slide 8 / 180 Slide 9 / 180 To take the square root of a variable rewrite its exponent as the square of a power. Square roots need to be positive answers. Even powered answered, like above, are positive even if the variables negative. The same cannot be said if the answer has an odd power. When you take a square root an the answer has an odd power, put the answer inside of absolute value signs.
Slide 10 / 180 Slide 11 / 180 Slide 12 / 180
Slide 13 / 180 Slide 14 / 180 Slide 15 / 180
Slide 16 / 180 10 C A B D no real solution Slide 17 / 180 11 C A B D no real solution Slide 18 / 180 12 C A B D no real solution
Slide 19 / 180 13 C A B D no real solution Slide 20 / 180 Slide 21 / 180 14 Evaluate A B C No Real Solution D
Slide 22 / 180 15 Evaluate A B C No Real Solution D Slide 23 / 180 16 Evaluate B A C No Real Solution D Slide 24 / 180
Slide 25 / 180 Intro to Cube Roots Return to Table of Contents Slide 26 / 180 Q: If a square root cancels a square, what cancels a cube? A: A cube root. Slide 27 / 180 The volume (V) of a cube is found by cubing its side length (s). V = s 3 V = s 3 V = 4 3 = 4 4 4 V = 64 cubic units or 64 units 3 The volume (V) of a cube is labeled as cubic units, or units 3 , because to find the 4 units volume, you need to cube its side.
Slide 28 / 180 A cube with sides 3 units would have a volume of 27 u 3 because 3 3 =27. If a cube has an volume of 64 u 3 what is the length of one side? Need to find a number when multiplied by itself three times will equal 64. 4 4 4 = 64, so 4 units is the length of a side. Slide 29 / 180 Slide 30 / 180
Slide 31 / 180 Slide 32 / 180 Slide 33 / 180
Slide 34 / 180 Slide 35 / 180 Slide 36 / 180
Slide 37 / 180 Slide 38 / 180 Slide 39 / 180 23 Simplify B A C D not possible
Slide 40 / 180 24 Simplify B A C D not possible Slide 41 / 180 Slide 42 / 180
Slide 43 / 180 27 Which of the following is not a step in simplifying A C D B Slide 44 / 180 nth Roots Return to Table of Contents Slide 45 / 180 In general, and absolute value signs are needed if n is even and the variable has an odd powered answer.
Slide 46 / 180 Slide 47 / 180 Slide 48 / 180 30 Simplify A B C D
Slide 49 / 180 31 Simplify A B C D Slide 50 / 180 Slide 51 / 180 33 Simplify A C D B
Slide 52 / 180 34 Simplify A B C D Slide 53 / 180 35 Simplify A B C D Slide 54 / 180 If the n th root of a radicand is , which of the following 36 is always true? No absolute value signs are ever needed. A Absolute value signs will always be needed. B Absolute value signs will be needed if j is negative. C Absolute value signs are needed if n is an even D index.
Slide 55 / 180 Rational Exponents Return to Table of Contents Slide 56 / 180 Rational Exponents, or exponents that are fractions, is another way to write a radical. Slide 57 / 180 Rewrite each radical as a rational exponent in the lowest terms.
Slide 58 / 180 Slide 59 / 180 Slide 60 / 180
Slide 61 / 180 39 Find the simplified expression that is equivalent to: A B C D Slide 62 / 180 40 Find the simplified expression that is equivalent to: A B C D Slide 63 / 180 41 Simplify A C B D
Slide 64 / 180 Slide 65 / 180 43 Simplify A B C D Slide 66 / 180 44 Simplify A B C D
Slide 67 / 180 Slide 68 / 180 45 Find the simplified expression that is equivalent to: A B C D Slide 69 / 180 46 Find the simplified expression that is equivalent to: A B C D
Slide 70 / 180 Simplifying Radicals is said to be a rational answer because their is a perfect square that equals the radicand. If a radicand doesn't have a perfect square that equals it, the root is said to be irrational . Slide 71 / 180 The square root of the following numbers is rational or irrational? Slide 72 / 180 The commonly excepted form of a radical is called the "simplified form". To simplify a non-perfect square, start by breaking the radicand into factors and then breaking the factors into factors and so on until there only prime numbers are left. this is called the prime factorization.
Slide 73 / 180 Slide 74 / 180 47 Which of the following is the prime factorization of 24? 3(8) A 4(6) B 2(2)(2)(3) C D 2(2)(2)(3)(3) Slide 75 / 180 48 Which of the following is the prime factorization of 72? 9(8) A B 2(2)(2)(2)(6) 2(2)(2)(3) C 2(2)(2)(3)(3) D
Slide 76 / 180 49 Which of the following is the prime factorization of 12? A 3(4) 2(6) B 2(2)(2)(3) C 2(2)(3) D Slide 77 / 180 50 Which of the following is the prime factorization of 24 rewritten as powers of factors? A B C D Slide 78 / 180 51 Which of the following is the prime factorization of 72 rewritten as powers of factors? A B C D
Slide 79 / 180 Slide 80 / 180 Slide 81 / 180 52 Simplify A B C D already in simplified form
Slide 82 / 180 53 Simplify A B C D already in simplified form Slide 83 / 180 54 Simplify A B C D already in simplified form Slide 84 / 180 55 Simplify A B C D already in simplified form
Slide 85 / 180 56 Which of the following does not have an irrational simplified form? A B C D Slide 86 / 180 Simplifying Roots of Variables Divide the index into the exponent. The number of times the index goes into the exponent becomes the power on the outside of the radical and the remainder is the power of the radicand. Slide 87 / 180 Simplifying Roots of Variables What about the absolute value signs? An Absolute Value sign is needed if the index is even, the starting power of the variable is even and the answer is an odd power on the outside. Examples of when absolute values are needed:
Slide 88 / 180 Slide 89 / 180 Slide 90 / 180
Slide 91 / 180 59 Simplify A B C D Slide 92 / 180 60 Simplify A B C D Slide 93 / 180 Operations with Radicals Return to Table of Contents
Slide 94 / 180 Addition and Subtraction Return to Table of Contents Slide 95 / 180 Operations with Radicals To add and subtract radicals they must be like terms. Radicals are like terms if they have the same radicands and the same indexes. Like Terms Unlike Terms Slide 96 / 180 61 Identify all of the pairs of like terms A B C D E F
Slide 97 / 180 To add or subtract radicals, only the coefficients of the like terms are combined. Slide 98 / 180 Slide 99 / 180 63 Simplify A B C Already Simplified D
Slide 100 / 180 Slide 101 / 180 65 Simplify A B C D Already Simplified Slide 102 / 180 66 Simplify A B C Already Simplified D
Slide 103 / 180 Some irrational radicals will not be like terms, but can be simplified. In theses cases, simplify then check for like terms. Slide 104 / 180 67 Simplify A B C D Already in simplest form Slide 105 / 180
Slide 106 / 180 Slide 107 / 180 70 Simplify A B C D Already in simplest form Slide 108 / 180 71 Which of the following expressions does not equal the other 3 expressions? A B C D
Slide 109 / 180 Multiplying Roots Return to Table of Contents Slide 110 / 180 Slide 111 / 180 72 Multiply A B C D
Slide 112 / 180 Multiplying Square Roots After multiplying, check to see if radicand can be simplified. Slide 113 / 180 73 Simplify A B C D Slide 114 / 180 74 Simplify A B C D
Slide 115 / 180 75 Simplify A B C D Slide 116 / 180 76 Simplify A B C D Slide 117 / 180 Multiplying Polynomials Involving Radicals 1) Follow the rules for distribution. 2)Be sure to simplify radicals when possible and combine like terms.
Slide 118 / 180 77 Multiply and write in simplest form: A B C D Slide 119 / 180 78 Multiply and write in simplest form: A B C D Slide 120 / 180 79 Multiply and write in simplest form: A B C D
Slide 121 / 180 80 Multiply and write in simplest form: A B C D Slide 122 / 180 81 Multiply and write in simplest form: A B C D Slide 123 / 180 Division: Rationalizing the Denominator Return to Table of Contents
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