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Copyright © 2007 Pearson Education, Inc. Slide R-2
Chapter R: Reference: Basic Algebraic Concepts
R.1 Review of Exponents and Polynomials
R.2 Review of Factoring
R.3 Review of Rational Expressions
R.4 Review of Negative and Rational Exponents
R.5 Review of Radicals
Copyright © 2007 Pearson Education, Inc. Slide R-3
R.5 Review of Radicals
Radical Notation for a1/n
If a is a real number, n is a positive integer, and a1/n is a real number, then
1/ .nn a a
Copyright © 2007 Pearson Education, Inc. Slide R-4
R.5 Review of Radicals
In the expression
• is called a radical sign,
• a is called the radicand,
• n is called the index.
n a
n
Copyright © 2007 Pearson Education, Inc. Slide R-5
R.5 Evaluating Roots
Example Evaluate each root.
(a) (b) (c)
Solution
(a)
(b) is not a real number.
(c)
4 16
1/ 4 4 1/ 44 16 16 (2 ) 2
4 16 5 32
5 1/55 32 [( 2) ] 2
4 16
Copyright © 2007 Pearson Education, Inc. Slide R-6
R.5 Review of Radicals
Radical Notation for am/n
If a is a real number, m is an integer, n is a positive integer, and is a real number, then
/ .m
nm n mna a a
n a
Copyright © 2007 Pearson Education, Inc. Slide R-7
R.5 Converting from Rational Exponents to Radicals
Example Write in radical form and simplify.
(a) (b) (c)
Solution
(a)
(b)
(c)
2/38 4/5( 32) 2/33x
22/3 238 8 2 4
44/5 45( 32) 32 ( 2) 16
32/3 23 3x x
Copyright © 2007 Pearson Education, Inc. Slide R-8
R.5 Converting from Radicals to Rational Exponents
Example Write in exponential form.
(a) (b) (c)
Solution
(a) (b)
(c)
54 ( 0)x x 2510 z 4 735 (2 )x
5 5/ 44 ( 0)x x x 22/5510 10z z
4 7 4 7 /3 7 /3 28/335 (2 ) 5(2 ) 5 2x x x
Copyright © 2007 Pearson Education, Inc. Slide R-9
R.5 Review of Radicals
Evaluating
If n is an even positive integer, then
If n is an odd positive integer, then
.n na a
n na
.n na a
Copyright © 2007 Pearson Education, Inc. Slide R-10
R.5 Using Absolute Value to Simplify Roots
Example Simplify each expression.
(a) (b) (c)
Solution
(a)
(b)
(c)
66 ( 2)44 p 8 616m r
44 p p
8 6 4 3 2 4 3 4 316 (4 ) 4 4m r m r m r m r
66 ( 2) 2 2
Copyright © 2007 Pearson Education, Inc. Slide R-11
R.5 Review of Radicals
Rules for Radicals
For all real numbers a and b, and positive integers m and n for which the indicated roots are real numbers,
( 0) .n
mn n n n mnnn
a aa b ab b a a
b b
Copyright © 2007 Pearson Education, Inc. Slide R-12
R.5 Using the Rules for Radicals to Simplify Radical Expressions
Example Simplify each expression.
(a) (b) (c)
Solution
(a)
(b)
(c)
7
646 54 3 23 m m
6 54 6 54 324 18
3 32 33 m m m m
7 7 7
64 864
Copyright © 2007 Pearson Education, Inc. Slide R-13
R.5 Simplifying Radicals
Simplified Radicals
An expression with radicals is simplified when the following conditions are satisfied.
1. The radicand has no factor raised to a power greater than or equal to the index.
2. The radicand has no fractions.
3. No denominator contains a radical.
4. Exponents in the radicand and the index of the radical have no common factor.
5. All indicated operations have been performed (if possible).
Copyright © 2007 Pearson Education, Inc. Slide R-14
R.5 Simplifying Radicals
Example Simplify each radical.
(a) (b)
Solution
(a)
(b)
175 5 7 63 81x y z
175 25 7 25 7 5 7
5 7 6 3 2 6 63 3
3 6 6 23
2 2 23
81 27 3
27 (3 )
3 3
x y z x x y y z
x y z x y
xy z x y
Copyright © 2007 Pearson Education, Inc. Slide R-15
R.5 Simplifying Radicals by Writing Them with Rational Exponents
Example Simplify each radical.
(a) (b)
Solution
(a)
(b)
6 23 12 36 ( 0)x y y
6 2 2/ 6 1/3 33 3 3 3
12 3 12 3 1/ 6 2 3/ 6 2 1/ 2 26 ( ) ( 0)x y x y x y x y x y y
Copyright © 2007 Pearson Education, Inc. Slide R-16
R.5 Adding and Subtracting Like Radicals
Example Add or subtract, as indicated. Assume all variables represent positive real numbers.
(a) (b)
Solution
(a)
7 2 8 18 4 72
7 2 8 18 4 72 7 2 8 9 2 4 36 2
7 2 8 3 2 4 6 2
7 2 24 2 24 2
7 2
398 3 32x y x xy
Copyright © 2007 Pearson Education, Inc. Slide R-17
R.5 Adding and Subtracting Like Radicals
Solution (b)
3 298 3 32 49 2 3 16 2
7 2 3 (4) 2
7 2 12 2
19 2
x y x xy x x y x x y
x xy x xy
x xy x xy
x xy
Copyright © 2007 Pearson Education, Inc. Slide R-18
R.5 Multiplying Radical Expressions
Example Find each product.
(a) (b)
Solution (a) Using FOIL,
2 3 8 5 7 10 7 10
2 3 8 5 2 8 2(5) 3 8 3(5)
16 5 2 3 2 2 15
4 5 2 6 2 15
11 2
Copyright © 2007 Pearson Education, Inc. Slide R-19
R.5 Multiplying Radical Expressions
Solution (b)
2 2
7 10 7 10 7 10
7 10
3
Copyright © 2007 Pearson Education, Inc. Slide R-20
R.5 Rationalizing Denominators
• The process of simplifying a radical expression so that no denominator contains a radical is called rationalizing the denominator.
• Rationalizing the denominator is accomplished by multiplying by a suitable form of 1.
Copyright © 2007 Pearson Education, Inc. Slide R-21
R.5 Rationalizing Denominators
Example Rationalize each denominator.
(a) (b)
Solution
(a)
(b)
4
3 3
2( 0)x
x
4 4 3 4 3
33 3 3
3 3 32 2 2
3 3 3 32 3
2 2 2 2x x x
xx x x x
Copyright © 2007 Pearson Education, Inc. Slide R-22
R.5 Rationalizing a Binomial Denominator
Example Rationalize the denominator of
Solution
1
1 2
1 1 21 1 21 2
1 21 2 1 2 1 2