Download 5.6A Rational Expressions

Chapter 5B Radical Functions
5.6A Simplifying Radical Expressions
Objective:
For the board: You will be able to rewrite radical expressions by using rational exponents.
You will be able to simplify and evaluate radical expressions and expressions containing
rational exponents.
Anticipatory Set:
The square of a number and the square root of a number are inverses.
Similarly the cube of a number and the cube root of a number are inverses.
Similarly the 4th, 5th, etc. power of a number and the 4th, 5th, etc. root of a number are inverses.
When taking a root, the 2nd, 3rd, 4th, etc. are called indexes.
Numbers and Types of Real Roots
Case
Odd index
Roots
1 real root
Even index;
positive radicand
Even index;
negative radicand
Radicand of 0
2 real roots
0 real roots
1 root
Example
The 3rd root of 8 is 2 because
23 = 8.
The 3rd root of -8 is -2 because
(-2)3 = -8.
The 4th roots of 16 are 2 and -2
because 24 = 16 and (-2)4 = 16.
-16 has no real 4th roots because
(no number)4 = -16.
The 3rd root of 0 is 0 because 03 = 0
Open the book to page 358 and read example 1.
Example: Find all real roots.
1. sixth roots of 64
Think (what)6 = 64.
Even index with a positive radicand so 2 roots.
2 and -2 because 26 = 64 and (-2)6 = 64
2. cube roots of -216
Think (what)3 = -216.
Odd index so 1 root.
-6 because (-6)3 = -216
3. fourth roots of -1024
Think (what)4 = -1024.
Even index with a negative radicand so no roots.
White Board Activity:
Find all real roots.
1. fourth roots of -256
None
2. sixth roots of 1
1 or -1
3. cube roots of 125
5
Instruction:
The nth root of a number a can be written as the radical expression n a where n is the index of the
radical and a is the radicand.
When a number has more than one real root, the radical sign indicates only the principal, or
positive root.
Properties of nth Roots
For a > 0 and b > 0,
Words
Product Property of Roots
The nth root of a product is equal
to the product of the nth roots.
Quotient Property of Roots
The nth root of a quotient is equal
to the quotient of the nth roots.
Numbers
3
16  3 8  3 2  23 2
3
3
64
64
4
3

125
125 5
Algebra
3
ab  3 a  3 b
3
a

b
3
a
3
b
Open the book to page 359 and read example 2.
Example: Simplify 4 81x12 . Assume that all variables are positive.
Prime factor the radicand and look for groups based on the index.
(3∙3∙3∙3)(xxxx)(xxxx)(xxxx) 3x3
White board activity:
Practice: Simplify 4 16x 4 each expression. Assume that all variables are positive.
(2∙2∙2∙2)(xxxx)
2x
Example: Simplify 3 16x 8
2x2
(2∙2∙2)(2)(xxx)(xxx)(xx)
3
2x 2
White Board Activity:
Practice: Simplify
3
24x10
(2·2·2)(3)(xxx)(xxx)(xxx)(x)
2x3 3 3x
To rationalize, (1) prime factor the numerator and denominator
(2) determine how many more of each factor is necessary to create the index number.
(3) multiply the numerator and denominator by that radical
(4) simplify.
Example: Simplify
2x 2
. Assume that all variables are positive.
5
To create a perfect 4th root, four 5’s are needed. Multiply the top and bottom of the
expression by 4 5  5  5
4
2x 2
4
5

2x 2
4
5
4
4
555
555

2 x 2 4 53
4
54

2 x 2 4 53
5
White board activity:
x8
. Assume that all variables are positive.
3
To create a perfect 4th root, four 3’s are needed. Multiply the top and bottom of the
expression by 4 3  3  3 .
Practice: Simplify
4
4
x 8 4 ( xxxx)( xxxx) 4 3  3  3 x 2 4 33
x 2 4 33



4
4
3
3
333
34
Assessment:
Question student pairs.
Independent Practice.
Text: pgs. 362 – 364 prob. 2 – 12, 30 – 40.
For a Grade:
Text: pgs. 352 – 264 prob. 9, 32, 38.