Post on 01-Apr-2015
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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)
Examples: Simplify each expression. Assume that all variables are positive.
125 20
2 12 3 27
2 2 3 338 25 8xy x y x y
544 32 2x x
R.8 nth Roots and Rational Expressions: Add, Subtract, Multiply and Divide Radicals
Multiplying and Dividing Radical Expressions:
Examples: Simplify each expression. Assume that all variables are positive.
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
R.8 nth Roots and Rational Expressions: Add, Subtract, Multiply and Divide Radicals
Examples: Simplify each expression. Assume that all variables are positive.
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.
Examples: Simplify each expression. Assume that all variables are positive.
24
5
5
4 2
3
4
2
R.8 nth Roots and Rational Expressions: Rationalize Denominators
Rationalizing Binomial Denominators:
example:
Examples: Simplify each expression. Assume that all variables are positive.
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
R.8 nth Roots and Rational Expressions: Simplify Expressions with Rational Exponents
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
R.8 nth Roots and Rational Expressions: Simplify Expressions with Rational Exponents
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
R.8 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational Exponents
Examples: Factor each expression. Express your answer so that only positive exponents occur.
4 13 32 24
4 4 23
x x x x
R.8 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational Exponents
Examples: Factor each expression. Express your answer so that only positive exponents occur.
312 26 2 3 8 0x x x x
R.8 nth Roots and Rational Expressions: Factor Expressions with Radicals/Rational Exponents
Examples: Factor each expression. Express your answer so that only positive exponents occur.
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
!
R.8 nth Roots and Rational Expressions
R.8 nth Roots and Rational Expressions
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