r.8 nth roots and rational expressions in this section, we will… evaluate nth roots simplify...

R.8 nth Roots and Rational Expressions

In this section, we will…

Evaluate nth Roots

Simplify Radical Expressions

Add, Subtract, Multiply and

Divide Radical Expressions

Rationalize Denominators

Simplify Expressions with Rational Exponents

Factor Expressions with Radicals or Rational Exponents

Chapter R Section 8: nth Roots and Rational Exponents

R.8 nth Roots and Rational Expressions: nth Roots

If a is a non-negative real number, any number b, such that is the square root of a and is denoted

If a is a non-negative real number, any non-negative number b, such that

is the primary square root of a and is denoted

Examples: Evaluate the following by taking the square root.

2b ab a

121

16

2b a b a

The principal root of a positive number is positive

Negative numbers do not have real # square roots

principal root:

Recall from Review Section 2…

R.8 nth Roots and Rational Expressions: nth Roots

The principal nth root of a real number a, n > 2 an integer, symbolized by

is defined as follows: where and if n is even where a, b are any real number if n is odd

Examples: Simplify each expression.3 27

0a 0b means nn a b a b n a

n aindexradicand

radical

3 8

4 81

4 16

5 1

principal root:

if 3 is oddn na a n

if 2 is evenn na a n

Properties of Radicals: Let and denote positive integers and let a and b represent real numbers. Assuming that all radicals are defined:

Simplifying Radicals: A radical is in simplest form when: No radicals appear in the denominator of a fraction The radicand cannot have any factors that are perfect roots

(given the index)

Examples: Simplify each expression.

2n 2m

n n nab a bn

nn

a a

b b m

n m na a

12

50

3 16

R.8 nth Roots and Rational Expressions: Simplify Radical Expressions

Simplifying Radical Expressions Containing Variables:

Examples: Simplify each expression. Assume that all variables are positive.

5 5x

84 16x

7b

6 53 54x y

When we divide the exponent by the index, the remainder remains

under the radical

R.8 nth Roots and Rational Expressions: Simplify Radical Expressions

R.8 nth Roots and Rational Expressions: Add, Subtract, Multiply and Divide Radicals

Adding and Subtracting Radical Expressions: simplify each radical expression combine all like-radicals

(combine the coefficients and keep the common radical)


125 20

2 12 3 27

2 2 3 338 25 8xy x y x y

544 32 2x x


Multiplying and Dividing Radical Expressions:


35 20x x

23

4 23

3

81

xy

x y

43 3 10

we will use: n n nab a b

we will use: n

naan

b b

we will use: m

n m na a



5 8 3 3

2 2x x

2

2 3 5

4 2 3 5 2 8

Vegas

Rule

R.8 nth Roots and Rational Expressions: Rationalize Denominators

Rationalizing Denominators: Recall that simplifying a radical expression means that no radicals appear in the denominator of a fraction.


24

5

5

4 2

3

4

2

R.8 nth Roots and Rational Expressions: Rationalize Denominators

Rationalizing Binomial Denominators:

example:


2

3 1

The conjugate of the binomial a + b is a – b and the conjugate of a – b is a + b.

2

3 1

2 1

3 2 2

R.8 nth Roots and Rational Expressions: Simplify Expressions with Rational Exponents

Evaluating Rational Exponents:

Examples: Simplify each expression.

If is a real number and 2 is an integer and assuming that all radicals are defined: a n

1416

180

If is a real number and and 2 are an integer and assuming that all radicals

are defined:

a m n

1n na a

mn

mn m na a a

324

3225

2327

8


Simplifying Expressions Containing Rational Exponents: Recall the following from Review Section 2:

Laws of Exponents: For any integers m, n (assuming no divisions by 0)m n m nx x x

nm mnx x

n n nxy x yn n

n

x x

y y

0 1x

mm n

n

xx

x

1n na a m

nm

n m na a a new!

new!

1nn

xx

n n

x y

y x

1 nn

xx and


Examples: Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive.

2 1 13 2 4x x x

344 8x y

1124

34

2 2

2

xy x y

x y

R.8 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational Exponents

Factoring Expressions with Radicals and/or Rational Exponents: Recall that, when factoring, we take out the GCF with the smallest exponent in the terms.

Examples: Factor each expression. Express your answer so that only positive exponents occur.

3 12 2x x

3 12 23 3x x

210 1 5 1x x x x



4 13 32 24

4 4 23

x x x x



312 26 2 3 8 0x x x x



4 13 322 3 4 4 3 4x x x x

R.8 nth Roots and Rational Expressions: Applications

Example: The final velocity, v, of an object in feet per second (ft/sec)

after it slides down a frictionless inclined plane of height h feet is:

where is the initial velocity

in ft/sec of the object.

What is the final velocity, v, of an object that slides down a frictionless inclined plane of height 2 feet with an initial velocity of 4 ft/sec?

2064v h v 0v

Independent Practice

You learn math by doing math. The best way to learn math is to practice, practice, practice. The assigned homework examples provide you with an opportunity to practice. Be sure to complete every assigned problem (or more if you need additional practice). Check your answers to the odd-numbered problems in the back of the text to see whether you have correctly solved each problem; rework all problems that are incorrect.

Read pp. 72-76

Homework:

pp. 77-79 #7-51 odds, 55-73 odds,

89-93 odds, 107Don

’t leave

your

grade

s to

chance

…do yo

ur ho

mework

!



From Math for Artists… “These are the laws of exponents and radicals in bright, cheerful, easy to memorize colors.”

Review of Exam Policies and Procedures

Page 7 of the Student Guide and Syllabus

r.8 nth roots and rational expressions in this section, we will… evaluate nth roots simplify...

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