Copyright © 2007 Pearson Education, Inc. Slide R-2 Chapter R: Reference: Basic Algebraic Concepts...

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



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



In the expression

• is called a radical sign,

• a is called the radicand,

• n is called the index.

n a

n


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



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


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


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



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


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



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


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


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).


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


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


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


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


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


R.5 Multiplying Radical Expressions

Solution (b)

2 2

7 10 7 10 7 10

7 10

3


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.


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


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

Copyright © 2007 Pearson Education, Inc. Slide R-2 Chapter R: Reference: Basic Algebraic Concepts...

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