hartfield – intermediate algebra (version 2014-2d) unit 4...
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Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 1
Topic 4‐1 Radical Expressions and Functions
What is a square root of 25? How many square roots does 25 have? Definition: X is a square root of a if X² = a. Symbolically, a is the principle square root of a. To symbolically represent each square root of a, one must write a and .a This leads to the short‐hand way of writing both square roots as .a
Do the following square roots exist? 4
4
0
4
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 2
What are the following square roots?
949
8
4
0.0016
64
x
z
In general….
is called a “radical sign” or a “root sign”. A square root is a particular type of root that uses the root sign for itself.
464z is an example of a radical expression since it an expression with a root sign. In the above expression, the 464z is the radicand. The radicand is the expression under (or better said, inside) a radical expression. ( )f x x is an example of a radical function.
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 3
Definition: A number S is called a perfect square if it’s the result of squaring an integer.
You need to memorize the first 21 numeric perfect squares. 0 1 121 4 144 9 169 16 196 25 225 36 256 49 289 64 324 81 361 100 400
The square root of a numeric value that isn’t a perfect square usually results in an irrational number. Recall that irrational numbers cannot be expressed as fractions of integers and their decimal form neither repeats nor terminates.
Variable expressions can be perfect squares also if we amend the definition as follows: An expression is a perfect square if its coefficient satisfies the definition of a numeric perfect square & each variable has an integer exponent that is a multiple of 2.
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 4
Definition: X is a cube root of a if X³ = a.
33 a X X a All numbers have one cube root thus every cube root is a principle cube root. 3
3
64
64
Definition: A number C is called a perfect cube if it’s the result of cubing an integer.
You need to memorize the first 11 numeric perfect cubes. 0 1 8 27 64 125 216 343 512 729 1000
Variable expressions can be perfect cubes also if we amend the definition as follows: An expression is a perfect cube if its coefficient satisfies the definition of a numeric perfect cube & each variable has an integer exponent that is a multiple of 3.
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 5
Definitions: X is a fourth root of a if X4 = a. X is a fifth root of a if X5 = a. X is an nth root of a if Xn = a. All roots have an index. The index of a root is equal to the power needed to return X to a by the previously state definitions. Roots with an even index (such as square roots and fourth roots)… Positive number have 2 real roots. Zero is its own root. Negative numbers have 0 real roots. Roots with an odd index (such as cube roots and fifth roots)… All numbers have exactly one real root.
Notationally write the fourth roots of 81 and evaluate.
Notationally write the fifth root of 243 and evaluate.
Definitions: A number R is called a perfect fourth if it’s the result of raising an integer to a fourth power. A number R is called a perfect fifth if it’s the result of raising an integer to a fifth power.
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 6
You need to memorize the first 6 numeric perfect fourths and first 5 numeric perfect fifths. Perfect fourths: 0 1 16 81 256 625
Perfect fifths: 0 1 32 243 1024 For roots with even indices, keep in mind the following rule:
If variables can represent any real number, you may need to use absolute value symbols when simplifying.
If the variables can only represent non‐negative numbers, you won’t need absolute value symbols when simplifying.
Absolute value symbols are never needed if a root has an odd index.
Find each root. Assume that all variables represent non‐negative real numbers.
4 4
9 123
12 2 6
625
125
64
x
x y
x y z
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Find each root. Assume that all variables can represent any real number.
4 4
4 124
4 8
5 20
625
32
x
x y
x
x
Find each root. Assume that all variables can represent any real number.
3 3
4 4
3 3
2
27
625
27
6 9
x
x
x
x x
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 8
Topic 4‐2 Radicals and Rational Exponents hw§1
Recall the Laws of Exponents (x > 0)
a b a b
aa b
b
n a na
x x x
xx
x
x x
Also: m m mxy x y and m m
mx xy y
Think about how the Laws of Exponents are related here:
2x x 12x x
33 x x 13 3x x
1n nx x
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 9
Explore the possibilities associated with the following rational exponent:
3
4x
In general we can conclude that
.
n mmn
mn
xx
x
Evaluate and/or simplify. Assume that all variables represent non‐negative real numbers.
12
23
18 2
112 3
49
8
9
64
x
x
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 10
Rewrite each expression in radical notation and simplify as possible. Assume that all variables represent non‐negative real numbers.
29 3
14 2
35
32 4
27
5
7
81
x
x
x
x
Recall that1mmxx
which we can extend to
define
1
.mn
mn
xx
Evaluate.
13
34
32
27
16
36
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 11
Use the properties of exponents to simplify each expression. Write your final answers with positive exponents.
4 13 2
23
34
x x
x
x
Use the properties of exponents to simplify each expression. Write your final answers with positive exponents.
456
415
11102
2
x
x
x x
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 12
Multiply. hw§2
1 12 23 2x x x
1 113 62 2 3 1x x x
Factor.
34
5 5x x
5 37 72x x
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 13
Use rational exponents with each to find a single simplified radical. Assume that all variables represent non‐negative real numbers.
4 2
9 36
8 64
x
x y
x
Use rational exponents with each to find a single simplified radical. Assume that all variables represent non‐negative real numbers.
4 3 6
5 2
x x
xx
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 14
Topic 4‐3 Product/Quotient Rules and Simplifying
Product Rule for Radicals: n n na b a b
Quotient Rule for Radicals: n
nn
a abb
Divide.
153
3
3542
Multiply. Assume that all variables represent non‐negative real numbers. 3 7 3 35 9x x 2 8x x 3 34 2
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 15
To simplify radicals, apply the product and quotient rules in reverse. Simplify. 75 162
Simplify.
48 3 128
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Simplify.
4 48 3 56
Emphasis: it’s all about perfect squares, cubes, etc. Simplify. Assume that all variables represent non‐negative real numbers.
5x 3 10x
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Simplify. Assume that all variables represent non‐negative real numbers.
318x 83 24y
Simplify. Assume that all variables represent non‐negative real numbers.
5 8 1150x y z 7 2 63 40x y z
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 18
Simplify. Assume that all variables represent non‐negative real numbers.
625
3 2
49x y
Simplify. Assume that all variables represent non‐negative real numbers.
412x
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 19
Topic 4‐4 Adding and Subtracting Radicals
Compare the following pairs of sums
2 3
3 4 3 2 4 3
3 4 3 2 4 2
3 4 3 2 4 2
x y
x x
x x
To add or subtract radicals, you must have like radicals. Like radicals have the same radicand and the same root index.
Add. 8 27 5 12
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 20
Subtract. 80 2 45
Subtract. 3 35 32 3 108
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Add and/or subtract. 4 50 3 24 5 32 54
Add. Assume that all variables represent non‐negative real numbers.
32 28 3 7x x x
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Add.
5 112 28
9 9
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 23
Topic 4‐5 More Multiplying Radicals
To multiply radicals with coefficients, keep the following rule in mind: n n nx a y b x y a b Multiply. 3 2 4 5 2
Multiply. 33 32 6 5 3 7 4
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Multiply. 7 2 3 4 6 2 3
Multiply. 2 6 3 2 2 6 3 2
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Multiply.
23 4 2
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 26
Topic 4‐6 Rationalizing Radical Expressions
Imagine if you had to divide the following expressions, which would be easier? (Note, 3 1.7320508 )
23 2 3
3
Traditionally, rationalizing the denominator of a radical expression was done for computational purposes. Today, it is used less frequently but is still a useful skill.
Rationalizing the denominator of a fraction:
to rewrite a fraction in an equivalent form where no radical is present in the denominator. There are three cases that vary the technique of rationalizing the denominator, based on what is in the denominator.
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 27
Case 1: The denominator is a square root.
Rationalize the denominator of the expression.
62
To rationalize the denominator when it is a square root, simplify the denominator and multiply by an identity fraction involving only the radical part of the simplified radical.
Rationalize the denominator of each expression.
512
2750
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 28
Rationalize the denominator of each expression.
156x
38
Case 2: The denominator is a cube root, fourth root, or any other root. Rationalize the denominator of the expression.
3
54
To rationalize the denominator when it is any root other than a square root, simplify the denominator, then determine the smaller perfect cube (fourth, etc) that the radicand of the denominator will divide. Create an identity fraction using an appropriate radical to create the perfect number under the root.
Hartfield – Intermediate Algebra (Version 2014-2D) Unit 4 | Page 29
Rationalize the denominator of each expression.
373
3
3
2 24 5
Rationalize the denominator of each expression.
3
336x
4 3
2
9x
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Case 3: The denominator is a square root ± a number or another root. Rationalize the denominator of the expression.
53 1
To rationalize the denominator when it consists of a square root plus or minus a number or another root, create an identity fraction using the conjugate pair of the denominator.
Rationalize the denominator of each expression.
3 47 3
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Rationalize the denominator of each expression.
4
7 3
Rationalize the denominator of each expression.
5
2 5