Slide 1 / 180. Radicals and Rational Exponents

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)

1 What is 1? Slide 5 / 180

2 What is? Slide 6 / 180

3 What is? Slide 7 / 180

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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.

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Slide 16 / 180 10 A C B D no real solution

Slide 17 / 180 11 A C B D no real solution

Slide 18 / 180 12 A C B D no real solution

Slide 19 / 180 13 A C B D no real solution

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Slide 21 / 180 14 Evaluate A B C D No Real Solution

Slide 22 / 180 15 Evaluate A B C D No Real Solution

Slide 23 / 180 16 Evaluate A B C D No Real Solution

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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 4 units The volume (V) of a cube is labeled as cubic units, or units 3, because to find the 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.

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Slide 39 / 180 23 Simplify A B C D not possible

Slide 40 / 180 24 Simplify A B C D not possible

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Slide 43 / 180 27 Which of the following is not a step in simplifying A C B D

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.

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Slide 48 / 180 30 Simplify A B C D

Slide 49 / 180 31 Simplify A B C D

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Slide 51 / 180 33 Simplify A B C D

Slide 52 / 180 34 Simplify A B C D

Slide 53 / 180 35 Simplify A B C D

Slide 54 / 180 36 If the n th root of a radicand is, which of the following is always true? A B C D No absolute value signs are ever needed. Absolute value signs will always be needed. Absolute value signs will be needed if j is negative. Absolute value signs are needed if n is an even 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.

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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

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Slide 65 / 180 43 Simplify A B C D

Slide 66 / 180 44 Simplify A B C D

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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.

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Slide 74 / 180 47 Which of the following is the prime factorization of 24? A 3(8) B 4(6) C 2(2)(2)(3) D 2(2)(2)(3)(3)

Slide 75 / 180 48 Which of the following is the prime factorization of 72? A 9(8) B 2(2)(2)(2)(6) C 2(2)(2)(3) D 2(2)(2)(3)(3)

Slide 76 / 180 49 Which of the following is the prime factorization of 12? A 3(4) B 2(6) C 2(2)(2)(3) D 2(2)(3)

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

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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:

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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.

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Slide 99 / 180 63 Simplify A B C D Already Simplified

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Slide 101 / 180 65 Simplify A B C D Already Simplified

Slide 102 / 180 66 Simplify A B C D Already Simplified

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

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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

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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

Slide 124 / 180 Rationalizing the Denominator Mathematicians don't like radicals in the denominators of fractions. When there is one, the denominator is said to be irrational. The method used to rid the denominator is termed "rationalizing the denominator". Which of these has a rational denominator? Rational Denominator Irrational Denominator

Slide 125 / 180 If a denominator needs to be rationalized, start by finding its conjugate. A conjugate is another polynomial that when the conjugate and the denominator are multiplied, no more irrational term. The conjugate for a monomial with a square root is the same square root. Example has a conjugate of. Why? Because The conjugate of a binomial with square roots is the opposite operation between the terms. Example has a conjugate of. Why? Because

Slide 126 / 180 Can you find a pattern for when a binomial is multiplied by its conjugate? Example Example Example Do you see a pattern that let's us go from line 1 to line 3 directly? (term 1) 2 - (term 2) 2

Slide 127 / 180 82 What is conjugate of? A B C D

Slide 128 / 180 83 What is conjugate of? A B C D

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Slide 130 / 180 85 What is conjugate of? A B C D

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Slide 132 / 180 The goal is to rationalize the denominator without changing the value of the fraction. To do this multiply the numerator and denominator by the same exact value. Examples:

Rationalize the Denominator: Slide 133 / 180 The original x in the radicand had an odd signs? power. Why no absolute value

Rationalize the Denominator: Slide 134 / 180

Slide 135 / 180 87 Simplify A B C D Already simplified

Slide 136 / 180 88 Simplify A B C D Already simplified

Slide 137 / 180 89 Simplify A B C D Already simplified

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Slide 140 / 180 Rationalizing n th roots of monomials Remember that, given an n th root in the denominator, you will need to find the conjugate that makes the radicand to the n th power. Examples:

Slide 141 / 180 92 Rationalize A B C D

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Slide 143 / 180 94 Rationalize A B C D

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Slide 145 / 180 Complex Numbers Return to Table of Contents

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Examples Slide 148 / 180

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Slide 150 / 180 97 Simplify A B C D

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Slide 153 / 180 100 Simplify A B C D

Slide 154 / 180 Higher order i's can be simplified down to a power of 1 to 4, which can be simplified into i, -1, -i, or 1. i i 2 i 3 i 4 i 5 =i 4 i i 6 = i 4 i 2 i 7 = i 4 i 3 i 8 = i 4 i 4 i 9 = i 4 i 4 i i 10 = i 4 i 4 i 2 i 11 = i 4 i 4 i 3 i 12 = i 4 i 4 i 4 i 13 = i 4 i 4 i 4 i i 14 = i 4 i 4 i 4 i 2 i 15 = i 4 i 4 i 4 i 3 i 16 = i 4 i 4 i 4 i 4............ i raised to a power can be rewritten as a product of i 4 's and an i to the 1 st to the 4 th. Since each i 4 = 1, we need only be concerned with the non-power of 4.

Slide 155 / 180 To simplify an i without writing out the table say i 87, divide by 4. The number of times 4 goes in evenly gives you that many i 4 's. The remainder is the reduced power. Simplify. Example: Simplify

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Slide 157 / 180 101 Simplify A i B -1 C -i D 1

Slide 158 / 180 102 Simplify A i B -1 C -i D 1

Slide 159 / 180 103 Simplify A i B -1 C -i D 1

Slide 160 / 180 104 Simplify A i B -1 C -i D 1

Slide 161 / 180 Complex Numbers Recall: Operations, such as addition and division, can be done with i. Treat i like any other variable, except at the end make sure i is at most to the first power. Use the following substitutions:

Examples: Slide 162 / 180

Slide 163 / 180 Examples (in the complex form the real term comes first)

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Slide 165 / 180 105 Simplify: A B C D

Slide 166 / 180 106 Simplify: A B C D

Slide 167 / 180 107 Simplify: A B C D

Slide 168 / 180 108 Simplify: A B C D

Slide 169 / 180 109 Simplify: A B C D

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Simplify Slide 171 / 180 Answers

Slide 172 / 180 110 Simplify A B C D

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Slide 174 / 180 112 Simplify A B C D

Slide 175 / 180 Dividing by i When dividing by a binomial with i, use the difference of squares to find the conjugate. Example: