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Radicalsand Rational Exponents
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Roots and RadicalsTable of Contents:
Square RootsIntro to Cube Rootsnth RootsIrrational RootsRational Exponents
Division: Rationalizing the Denominator
Operations with Radicals
Addition and Subtraction
Multiplication
Complex Numbers
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Roots
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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 (42=16 and (-4)2=16)
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1 What is 1 ?
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2 What is ?
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3 What is ?
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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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10
A
B
C
D no real solution
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11
A
B
C
D no real solution
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12
A
B
C
D no real solution
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13
A
B
C
D no real solution
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14 Evaluate
A B
C D No Real Solution
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15 Evaluate
A B
C D No Real Solution
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16 Evaluate
A B
C D No Real Solution
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Intro to Cube Roots
Return to Tableof Contents
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Q: If a square root cancels a square, what cancels a cube?
A: A cube root.
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The volume (V) of a cube is found by cubing its side length (s).
V = s3
V = s3
V = 43 = 4 4 4
V = 64 cubic units or 64 units3
4 units
The volume (V) of a cube is labeled as cubic units, or units3, because to find the volume, you need to cube its side.
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A cube with sides 3 units would have a volume of 27 u3 because 33=27.
If a cube has an volume of 64 u3 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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23 Simplify
A B
C D not possible
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24 Simplify
A B
C D not possible
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27 Which of the following is not a step in simplifying
A
B
C
D
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nth Roots
Return to Tableof Contents
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In general, and absolute value signs are needed if n is even and the variable has an odd powered answer.
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30 Simplify
A
B
C
D
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31 Simplify
A
B
C
D
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33 Simplify
A
B
C
D
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34 Simplify
A
B
C
D
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35 Simplify
A
B
C
D
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36 If the nth root of a radicand is , which of the following is always true?
A No absolute value signs are ever needed.
B Absolute value signs will always be needed.
C Absolute value signs will be needed if j is negative.
D Absolute value signs are needed if n is an even index.
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Rational Exponents
Return to Tableof Contents
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Rational Exponents, or exponents that are fractions, is another way to write a radical.
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Rewrite each radical as a rational exponent in the lowest terms.
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39 Find the simplified expression that is equivalent to:
A
B
C
D
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40 Find the simplified expression that is equivalent to:
A
B
C
D
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41 Simplify
A
B
C
D
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43 Simplify
A
B
C
D
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44 Simplify
A
B
C
D
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45 Find the simplified expression that is equivalent to:
A
B
C
D
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46 Find the simplified expression that is equivalent to:
A
B
C
D
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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.
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The square root of the following numbers is rational or irrational?
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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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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)
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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)
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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)
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50 Which of the following is the prime factorization of 24 rewritten as powers of factors?
A
B
C
D
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51 Which of the following is the prime factorization of 72 rewritten as powers of factors?
A
B
C
D
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52 Simplify
A
B
C
D already in simplified form
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53 Simplify
A
B
C
D already in simplified form
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54 Simplify
A
B
C
D already in simplified form
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55 Simplify
A
B
C
D already in simplified form
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56 Which of the following does not have an irrational simplified form?
A
B
C
D
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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.
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Simplifying Roots of VariablesWhat 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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59 Simplify
A
B
C
D
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60 Simplify
A
B
C
D
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Operations with Radicals
Return to Tableof Contents
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Addition and Subtraction
Return to Tableof Contents
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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
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61 Identify all of the pairs of like terms
A
B
C
D
E
F
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To add or subtract radicals, only the coefficients of the like terms are combined.
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63 Simplify
A
B
C
D Already Simplified
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65 Simplify
A
B
C
D Already Simplified
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66 Simplify
A
B
C
D Already Simplified
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Some irrational radicals will not be like terms, but can be simplified. In theses cases, simplify then check for like terms.
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67 Simplify
A
B
C
D Already in simplest form
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70 Simplify
A
B
C
D Already in simplest form
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71 Which of the following expressions does not equal the other 3 expressions?
A
B
C
D
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Multiplying Roots
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72 Multiply
A
B
C
D
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Multiplying Square Roots
After multiplying, check to see if radicand can be simplified.
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73 Simplify
A
B
C
D
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74 Simplify
A
B
C
D
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75 Simplify
A
B
C
D
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76 Simplify
A
B
C
D
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Multiplying Polynomials Involving Radicals 1) Follow the rules for distribution. 2)Be sure to simplify radicals when possible and combine like terms.
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77 Multiply and write in simplest form:
A
B
C
D
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78 Multiply and write in simplest form:
A
B
C
D
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79 Multiply and write in simplest form:
A
B
C
D
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80 Multiply and write in simplest form:
A
B
C
D
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81 Multiply and write in simplest form:
A
B
C
D
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Division: Rationalizing the Denominator
Return to Tableof Contents
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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?
RationalDenominator
IrrationalDenominator
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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
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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
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82 What is conjugate of ?
A
B
C
D
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83 What is conjugate of ?
A
B
C
D
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85 What is conjugate of ?
A
B
C
D
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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:
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Rationalize the Denominator:
The original x in the radicand had an odd
power.
Why no absolute value signs?
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Rationalize the Denominator:
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87 Simplify
A
B
C
D Already simplified
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88 Simplify
A
B
C
D Already simplified
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89 Simplify
A
B
C
D Already simplified
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Rationalizing nth roots of monomials
Remember that , given an nth root in the denominator, you will need to find the conjugate that makes the radicand to the nth power.
Examples:
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92 Rationalize
A
B
C
D
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94 Rationalize
A
B
CD
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Examples
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97 Simplify
A
B
C
D
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100 Simplify
A
B
C
D
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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 i2 i3 i4
i5 =i4 i i6 = i4 i2 i7 = i4 i3 i8 = i4 i4
i9 = i4 i4 i i10 = i4 i4 i2 i11 = i4 i4 i3 i12 = i4 i4 i4
i13 = i4 i4 i4 i i14 = i4 i4 i4 i2 i15 = i4 i4 i4 i3 i16 = i4 i4 i4 i4
... ... ... ...
i raised to a power can be rewritten as a product of i4 's and an i to the 1st to the 4th.
Since each i4 = 1, we need only be concerned with the non-power of 4.
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To simplify an i without writing out the table say i87, divide by 4.
The number of times 4 goes in evenly gives you that many i4 's.The remainder is the reduced power. Simplify.
Example: Simplify
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101 Simplify
A i
B -1
C -i
D 1
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102 Simplify
A i
B -1
C -i
D 1
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103 Simplify
A i
B -1
C -i
D 1
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104 Simplify
A i
B -1
C -i
D 1
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Complex Numbers
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:
Recall:
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Examples:
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Examples (in the complex form the real term comes first)
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105 Simplify:
A
B
C
D
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106 Simplify:
A
B
C
D
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107 Simplify:
A
B
C
D
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108 Simplify:
A
B
C
D
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109 Simplify:
A
B
C
D
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Simplify Answers
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110 Simplify
A
B
C
D
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112 Simplify
A
B
C
D
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Dividing by iWhen dividing by a binomial with i, use the difference of squares to find the conjugate.
Example:
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Simplify: Answers
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114 Simplify:
A
B
C
D
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115 Simplify:
A
B
C
D
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116 Simplify:
A
B
C
D
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