Download Topic 4-1 Radical Expressions and Functions What is a square root

Hartfield – Intermediate Algebra
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 . Unit 4
(Version 2015 - © Thomas Hartfield)
Do the following square roots exist? 4
 4
0
4
Hartfield – Intermediate Algebra
What are the following square roots? 9
49
0.0016
 x8
64 z 4
(Version 2015 - © Thomas Hartfield)
Unit 4
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. 64z 4 is an example of a radical expression since it an expression with a root sign. In the above expression, the 64z 4 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
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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 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 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. Unit 4
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. Hartfield – Intermediate Algebra
Definition: X is a cube root of a if X³ = a. 3
a  X  X3  a All numbers have one cube root thus every cube root is a principle cube root. 3
64
3
64
Unit 4
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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 2015 - © Thomas Hartfield)
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. Unit 4
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
Unit 4
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You need to memorize the first 6 numeric perfect fourths and first 5 numeric perfect fifths. Perfect fourths: 0 1 16 81 256 625 Find each root. Assume that all variables represent non‐negative real numbers. 4
625x 4
3
125x 9 y 12 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. 64 x 12 y 2 z 6
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. Hartfield – Intermediate Algebra
Find each root. Assume that all variables can represent any real number. 625x 4
3
27 x 3
4
x 4 y 12
4
625x 4
4
x8
3
27 x 3
5
Find each root. Assume that all variables can represent any real number. 4
Unit 4
(Version 2015 - © Thomas Hartfield)
x2  6 x  9
32 x 20
Hartfield – Intermediate Algebra
Topic 4‐2 Radicals and Rational Exponents Recall the Laws of Exponents (x > 0) x x  x
a
b
xa
xb
 
xa
n
ab
 x a b Also:  xy 
hw§1
Think about how the Laws of Exponents are related here:  x
 x a n
m
m
xm
x
 x  y and    m y
y
m
m
Unit 4
(Version 2015 - © Thomas Hartfield)
2
 x
3
3
 x  x 1
x  x 2 3
n
1
x  x 3 1
x  x n Hartfield – Intermediate Algebra
Explore the possibilities associated with the following rational exponent: x
3
Evaluate and/or simplify. Assume that all variables represent non‐negative real numbers. 4 In general we can conclude that x
Unit 4
(Version 2015 - © Thomas Hartfield)
49
8
2
1
2
3
9x 
8
m
n


n
xm
 x
n
1
2
m . 
64 x 12

1
3
Hartfield – Intermediate Algebra
Rewrite each expression in radical notation and simplify as possible. Assume that all variables represent non‐
negative real numbers. 27x 
9
 
5x 4
7x
3
define x
m
Evaluate. 3
27
2
3
4
n
m

1

x
16
1
3
3
4 5
81x 
Recall that x
2
1
2
Unit 4
(Version 2015 - © Thomas Hartfield)
36
3
2
m
1
x
m
. n
which we can extend to Hartfield – Intermediate Algebra
Use the properties of exponents to simplify each expression. Write your final answers with positive exponents. x
x
x
4
3
2
3
x
1
Use the properties of exponents to simplify each expression. Write your final answers with positive exponents. 
x

2
3
4
Unit 4
(Version 2015 - © Thomas Hartfield)
5
6



4
4
 15 
 2x  

x
1
2
x
1
10
Hartfield – Intermediate Algebra
Multiply. 1 
1 
2
x  3x  2 x 2  

1 
1
 12
3
6  1 
x
2
x
3
x






Unit 4
(Version 2015 - © Thomas Hartfield)
hw§2
Factor. x
4
5
3
 x 5 x
5
7
3
 2x 7 Hartfield – Intermediate Algebra
Use rational exponents with each to find a single simplified radical. Assume that all variables represent non‐negative real numbers. 4
x2
6
x9y3
8
4 x6
Unit 4
(Version 2015 - © Thomas Hartfield)
Use rational exponents with each to find a single simplified radical. Assume that all variables represent non‐negative real numbers. 4
x3  6 x
5
x2
x
Hartfield – Intermediate Algebra
(Version 2015 - © Thomas Hartfield)
Topic 4‐3 Product/Quotient Rules and Simplifying Product Rule for Radicals: a  n b  n ab n
n
Quotient Rule for Radicals: Divide. 15
3
n
3
54
3
2
a
b

n
a
b
Multiply. Assume that all variables represent non‐
negative real numbers. 3
3  7 5x  3 9 x 3
2x  8x 4 3 2 Unit 4
Hartfield – Intermediate Algebra
To simplify radicals, apply the product and quotient rules in reverse. Simplify. 75 Unit 4
(Version 2015 - © Thomas Hartfield)
162 Simplify. 48 3
128 Hartfield – Intermediate Algebra
Simplify. 4
48 Unit 4
(Version 2015 - © Thomas Hartfield)
Emphasis: it’s all about perfect squares, cubes, etc. 3
56 Simplify. Assume that all variables represent non‐
negative real numbers. x5 3
x 10 Hartfield – Intermediate Algebra
Simplify. Assume that all variables represent non‐
negative real numbers. 18x 3 Unit 4
(Version 2015 - © Thomas Hartfield)
3
Simplify. Assume that all variables represent non‐
negative real numbers. 24y 8 50x 5 y 8 z11 3
40x 7 y 2 z 6 Hartfield – Intermediate Algebra
(Version 2015 - © Thomas Hartfield)
Simplify. Assume that all variables represent non‐
negative real numbers. 6
25
x 3y2
49
Simplify. Assume that all variables represent non‐
negative real numbers. 12
4 x
Unit 4
Hartfield – Intermediate Algebra
Topic 4‐4 Adding and Subtracting Radicals Compare the following pairs of sums 3x  4 y
3 2 4 3
3x  4 x 2
3 2  43 2
3x  4 x
3 2 4 2
To add or subtract radicals, you must have like radicals. Like radicals have the same radicand and the same root index. Unit 4
(Version 2015 - © Thomas Hartfield)
Add. 8 27  5 12 Hartfield – Intermediate Algebra
Subtract. 80  2 45 (Version 2015 - © Thomas Hartfield)
Subtract. 5 3 32  3 3 108 Unit 4
Hartfield – Intermediate Algebra
Add and/or subtract. 4 50  3 24  5 32  54 (Version 2015 - © Thomas Hartfield)
Unit 4
Add. Assume that all variables represent non‐negative real numbers. 2 x 28 x  3 7 x 3 Hartfield – Intermediate Algebra
Add. 5 112
28

9
9
(Version 2015 - © Thomas Hartfield)
Unit 4
Hartfield – Intermediate Algebra
Topic 4‐5 More Multiplying Radicals To multiply radicals with coefficients, keep the following rule in mind: xn a yn b  x y n ab Multiply. 3 2 4 5 2 

Unit 4
(Version 2015 - © Thomas Hartfield)
Multiply. 23 6 5  3 3  73 4 

Hartfield – Intermediate Algebra
Multiply. 7 2  3 4 6 2 3 

Unit 4
(Version 2015 - © Thomas Hartfield)

Multiply. 2 6 3 2 2 6 3 2 


Hartfield – Intermediate Algebra
Multiply. 

2
34 2 (Version 2015 - © Thomas Hartfield)
Unit 4
Hartfield – Intermediate Algebra
(Version 2015 - © Thomas Hartfield)
Topic 4‐6 Rationalizing Radical Expressions Rationalizing the denominator of a fraction: Imagine if you had to divide the following expressions, which would be easier? (Note, 3  1.7320508 ) 2
2 3
3
3
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. 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. Unit 4
Hartfield – Intermediate Algebra
(Version 2015 - © Thomas Hartfield)
Case 1: The denominator is a square root. Rationalize the denominator of the expression. 6
2
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. 5
27
12
50
Unit 4
Hartfield – Intermediate Algebra
Rationalize the denominator of each expression. 15
3
8
6x
(Version 2015 - © Thomas Hartfield)
Unit 4
Case 2: The denominator is a cube root, fourth root, or any other root. Rationalize the denominator of the expression. 5
3
4
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
Rationalize the denominator of each expression. 3
7
2
2
3
3
3
4 5
(Version 2015 - © Thomas Hartfield)
Rationalize the denominator of each expression. 3
2
3
4
3
36 x
9x
Unit 4
Hartfield – Intermediate Algebra
(Version 2015 - © Thomas Hartfield)
Case 3: The denominator is a square root ± a number or another root. Rationalize the denominator of the expression. 5
3 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4
7 3
Unit 4
Hartfield – Intermediate Algebra
Rationalize the denominator of each expression. 4
7 3
(Version 2015 - © Thomas Hartfield)
Rationalize the denominator of each expression. 5
2 5
Unit 4