chapter 7 radicals, radical functions, and rational exponents

Chapter 7Radicals, Radical

Functions, and Rational Exponents

§ 7.1

Radical Expressions and Functions

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.1

RadicalsIn this section, we introduce a new category of expressions and functionsthat contain roots.

For example, the reverse operation of squaring a number is finding thesquare root of the number.

The symbol that we use to denote the principal square root

is called a radical sign. The number under the radical sign is called the radicand. Together we refer to the radical sign and its radicand as aradical expression.


Radical ExpressionsEXAMPLE

n a

Radical Expression

Radicand

Radical Sign

Index of the Radical


Radical Expressions

Definition of the Principal Square RootIf a is a nonnegative real number, the nonnegative number b such that , denoted by , is the principal square root of a.

ab 2 ab

P 487



Evaluate: .25144(c)25144(b)16(a)

SOLUTIONThe principal square root of a negative number, -16, is not a real number.Simplify the radicand. The principal square root of 169 is 13.

16(a)

1316925144(b)

1751225144(c) Take the principal square root of 144, 12, and of 25, 5, and then add to get 17.


Radical ExpressionsCheck Point 1 on page 487

Evaluate:64

49-

525169 743169

87

2516

54

0081.0 09.0

Principal Square Root means the answer is nonnegative

- Denotes the negative square root of a number

Is a grouping symbol


Radical Expressions

Square Root FunctionBecause each nonnegative real number, x, has precisely one principal square root, , there is a square root function defined by The domain of this function is .

xxf )(

x

Bottom of P 487

),0[

x (x,y)0 (0,0)

1 (1,1)

4 (4,2)

9 (9,3)

16 (16,4)


Radical Expressions

xxf )(

P 488

00)0( f

11)1( f

24)4( f

39)9( f

416)16( f

See Figure 7.1 on page 488


Radical FunctionsEXAMPLE

For the function, find the indicated function value:

.1,21,1,4;12

ggggxxg

SOLUTION

Substitute 4 for x in 39

1424

g

73.13

1121

g

.12 xxg

Simplify the radicand and take the square root of 9.

Substitute 1 for x in .12 xxg

Simplify the radicand and take the square root of 3.


Radical Functions

Substitute -1/2 for x in

00

1212

21

g

1

1121g

.12 xxgSimplify the radicand and take the square root.

Substitute -1 for x in .12 xxgSimplify the radicand. The principal square root of a negative number is not a real number.

CONTINUED


Radical FunctionsCheck Point 2


xxgxxf 39;2012

SOLUTION

3f

5g

4

90.424

P 488

20312

)5(39


Radical Functions - Domain

We have seen that the domain of a function f is the largest set of real numbers for which the value of f(x) is defined.

Because only nonnegative numbers have real square roots, the domain of a square root function is the set of real numbers for which the radicand is nonnegative.

In other words, we only use “allowable” x in the domain of the function. Not allowed for x is any value of x that would cause a negative number under a square root.


Radical Functions - DomainEXAMPLE

Find the domain of .153 xxf

SOLUTIONThe domain is the set of real numbers, x, for which the radicand, 3x – 15, is nonnegative. We set the radicand greater than or equal to 0 and solve the resulting inequality.

The domain of f is . ,5or 5| xx

0153 x153 x5x



Find the Domain of

279 xxf

SOLUTION

),3[or 3| : xxxfdomain

P 489


Radical Functions in ApplicationEXAMPLE

Police use the function to estimate the speed of a car, f (x), in miles per hour, based on the length, x, in feet, of its skid marks upon sudden braking on a dry asphalt road. Use the function to solve the following problem.

A motorist is involved in an accident. A police officer measures the car’s skid marks to be 45 feet long. Estimate the speed at which the motorist was traveling before braking. If the posted speed limit is 35 miles per hour and the motorist tells the officer she was not speeding, should the officer believe her? Explain.

xxf 20


Radical Functions in Application

SOLUTION

Use the given function. xxf 20

CONTINUED

Substitute 45 for x. 4520xf

Simplify the radicand. 900xf

Take the square root. 30xf

The model indicates that the motorist was traveling at 30 miles per hour at the time of the sudden braking. Since the posted speed limit was 35 miles per hour, the officer should believe that she was not speeding.


Radical Expressions

Simplifying TFor any real number a,

In words, the principal square root of is the absolute value of a.

.2 aa

2a

2a

The principal root is the positive root. P 490

(a) To simplify , first write as an expression that is squared: . Then simplify.



Simplify each expression:

.4914(b)81(a) 24 xxx

SOLUTIONThe principal square root of an expression squared is the absolute value of that expression. In both exercises, it will first be necessary to express the radicand as an expression that is squared.

481x 481x 224 981 xx

22224 9or 9981 xxxx

(b) To simplify , first write as an expression that is squared: . Then simplify.


Radical Expressions

49142 xx

CONTINUED

49142 xx 22 74914 xxx

774914 22 xxxx



Simplify each expression:

7

8 x

P 490

2)7( . a

2)8( . xb

1049 . xc57x

96 . 2 xxd 3 x


Radical Expressions

Definition of the Cube Root of a NumberThe cube root of a real number a is written .

. that means 33 abba

3 a

P 491


Radical FunctionsEXAMPLE


.63,0,13;123 gggxxg

SOLUTION

Substitute 13 for x in 327

1132133

3

g

11

1020 3

g

.123 xxg

Simplify the radicand and take the cube root of 27.

Substitute 0 for x in .123 xxg

Simplify the radicand and take the cube root of 1.


Radical Functions

Substitute -63 for x in 55125

1632633

3

g .123 xxg

Simplify the radicand and take the cube root of -125 and then simplify.

CONTINUED

x (x,y)-8 (-8,-2)

-1 (-1,-1)

0 (0,0)

1 (1,1)

8 (8,2)


Radical Expressions

3)( xxf

P 491

28)8( 3 f

See Figure 7.2 on page 491

11)1( 3 f

00)0( 3 f

11)1( 3 f

28)8( 3 f




3 6 . xxfa

33f

5g

3

2

P 492

3 633

3 2)5(2

3 22 . xxgb

3 27

3 8


Radical Expressions


In words, the cube root of any expression is that expression cubed.

.3 3 aa

3 3a



Simplify: .1253 3x

SOLUTION

Begin by expressing the radicand as an expression that is cubed: . Then simplify.

We can check our answer by cubing -5x:

33 5125 xx

xxx 55125 3 33 3

3333 12555 xxx

By obtaining the original radicand, we know that our simplification is correct.



Simplify:

x3

P 492

3 327x



Find the indicated root, or state that the expression is not a real number:

.1(b)1 (a) 85

SOLUTION

is not a real number because the index, 8, is even and the radicand, -1, is negative. No real number can be raised to the eighth power to give a negative result such as -1. Real numbers to even powers can only result in nonnegative numbers.

because . An odd root of a negative real number is always negative.

11 (a) 5 1111111 5

8 1(b)


Radical Expressions


1) If n is even,

2) If n is odd,. aan n

n na

. aan n



Simplify: .232(b)5(a) 5 54 4 xx

SOLUTIONEach expression involves the nth root of a radicand raised to the nth power. Thus, each radical expression can be simplified. Absolute value bars are necessary in part (a) because the index, n, is even.

if n is even. 55(a) 4 4 xx aan n

2222

22232(b)

5 5

5 555 5

xx

xx if n is odd.aan n



Find the indicated root, or state not real:

2

2

P 493

4 16 .a

4 16- .b

4 16 . c realnot

5 1- .d 1



Simply:

6 x

P 494

4 46 . xa

6 6)8( . c 8

5 52-3x .b 23 x

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