Topic 4 1 Radical Expressions Functions Do root of How ...faculty.ung.edu/thartfield/courses/2015-1-spring/0099_notes_unit4... · fractions of integers and their decimal form neither

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Hartfield – Intermediate Algebra (Version 2015 - © Thomas Hartfield) Unit 4

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


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.


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.


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.


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.


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


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


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


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


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


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


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


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


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


To simplify radicals, apply the product and quotient rules in reverse. Simplify. 75 162

Simplify.

48 3 128


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


Simplify. Assume that all variables represent non‐negative real numbers.

318x 83 24y


5 8 1150x y z 7 2 63 40x y z



625

3 2

49x y


412x


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


Subtract. 80 2 45

Subtract. 3 35 32 3 108


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


Add.

5 112 28

9 9


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


Multiply. 7 2 3 4 6 2 3

Multiply. 2 6 3 2 2 6 3 2


Multiply.

23 4 2


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.


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



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.



373

3

3

2 24 5


3

336x

4 3

2

9x


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.


3 47 3



4

7 3


5

2 5

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Topic 4 1 Radical Expressions Functions Do root of How ...faculty.ung.edu/thartfield/courses/2015-1-spring/0099_notes_unit4... · fractions of integers and their decimal form neither