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Chapter 7 Rational Exponents, Radicals, and Complex Numbers

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Chapter 7. Rational Exponents, Radicals, and Complex Numbers. Radicals and Radical Functions. § 7.1. Square Roots. Opposite of squaring a number is taking the square root of a number. A number b is a square root of a number a if b 2 = a. - PowerPoint PPT Presentation

### Transcript of Rational Exponents, Radicals, and Complex Numbers

Chapter 7

Rational Exponents, Radicals, and

Complex Numbers

§ 7.1

Martin-Gay, Intermediate Algebra, 5ed 33

Square Roots

Opposite of squaring a number is taking the square root of a number.

A number b is a square root of a number a if b2 = a.

In order to find a square root of a, you need a # that, when squared, equals a.

Martin-Gay, Intermediate Algebra, 5ed 44

Principal and Negative Square Roots

If a is a nonnegative number, then

is the principal or nonnegative square root of a

a

is the negative square root of a.a

Principal Square Roots

Martin-Gay, Intermediate Algebra, 5ed 55

Radical expression is an expression containing a radical sign.

Radicand is the expression under a radical sign.

Note that if the radicand of a square root is a negative number, the radical is NOT a real number.

Martin-Gay, Intermediate Algebra, 5ed 66

49 7

16

25

4

5

4 2

Example:

Martin-Gay, Intermediate Algebra, 5ed 77

Square roots of perfect square radicands simplify to rational numbers (numbers that can be written as a quotient of integers).

Square roots of numbers that are not perfect squares (like 7, 10, etc.) are irrational numbers.

IF REQUESTED, you can find a decimal approximation for these irrational numbers.

Otherwise, leave them in radical form.

Perfect Squares

Martin-Gay, Intermediate Algebra, 5ed 88

Radicands might also contain variables and powers of variables.

To avoid negative radicands, assume for this chapter that if a variable appears in the radicand, it represents positive numbers only.

1064x 58x

Perfect Square Roots

Example:

Martin-Gay, Intermediate Algebra, 5ed 99

Cube Root

The cube root of a real number a is written as

3 , anda

Cube Roots

abba 33 ifonly

Martin-Gay, Intermediate Algebra, 5ed 1010

3 27 3

3 68x 22x

Cube Roots

Example:

Martin-Gay, Intermediate Algebra, 5ed 1111

Other roots can be found, as well.

The nth root of a is defined as

abba nn ifonly

If the index, n, is even, the root is NOT a real number when a is negative.

If the index is odd, the root will be a real number.

nth Roots

Martin-Gay, Intermediate Algebra, 5ed 1212

Simplify the following.

20225 ba 105ab

39

364

b

a3

4

b

a

nth Roots

Example:

Martin-Gay, Intermediate Algebra, 5ed 1313

nth Roots

Simplify the following. Assume that all variables represent positive numbers.

4 816x 22x

515

532

b

a

3

2

b

a3

2

b

a

Example:

Martin-Gay, Intermediate Algebra, 5ed 1414

If the index of the root is even, then the notation represents a positive number.

n a

But we may not know whether the variable a is a positive or negative value.

Since the positive square root must indeed be positive, we might have to use absolute value signs to guarantee the answer is positive.

nth Roots

Martin-Gay, Intermediate Algebra, 5ed 1515

If n is an even positive integer, then

aan n

If n is an odd positive integer, then

aan n

Finding nth Roots

Finding n na

Martin-Gay, Intermediate Algebra, 5ed 1616

Simplify the following.

20225 ba 105ab 105 ba

If we know for sure that the variables represent positive numbers, we can write our result without the absolute value sign.

10202 525 abba

Finding nth Roots

Martin-Gay, Intermediate Algebra, 5ed 1717

39

364

b

a3

4

b

a

Simplify the following.

Since the index is odd, we don’t have to force the negative root to be a negative number.

If a or b is negative (and thus changes the sign of the answer), that’s okay.

Finding nth Roots

Example:

Martin-Gay, Intermediate Algebra, 5ed 1818

We can also use function notation to represent rational functions.

For example,

Evaluating a rational function for a particular value involves replacing the value for the variable(s) involved.

Evaluating Rational Functions

Example:

2( ) 4.xT xx

Find the value 2(2) 4.xT

x

222

) 42(T 4 42

2 4 2

Martin-Gay, Intermediate Algebra, 5ed 1919

Since every value of x that is substituted into the equation

n xy

produces a unique value of y, the root relation actually represents a function.

The domain of the root function when the index is even, is all nonnegative numbers.

The domain of the root function when the index is odd, is the set of all real numbers.

Root Functions

Martin-Gay, Intermediate Algebra, 5ed 2020

We have previously worked with graphing basic forms of functions so that you have some familiarity with their general shape.

You should have a basic familiarity with root functions, as well.

Root Functions

Martin-Gay, Intermediate Algebra, 5ed 2121

x y

4 2

1 1

0 0

x

y

(0, 0)

(4, 2)

(1, 1)

Graph xy

6 6

2 2

(2, )2(6, )6

Graphs of Root Functions

Example:

Martin-Gay, Intermediate Algebra, 5ed 2222

x y

1 1

0 0x

y

(0, 0)

(1, 1)

28

4 3 4 (4, )3 4(8, 2)

-1 -1

-4 3 4

-2-8

(-1, -1)

(-4, )3 4(-8, -2)

Graph 3 xy

Graphs of Root Functions

Example:

Martin-Gay, Intermediate Algebra, 5ed 2323

§ 7.2

Rational Exponents

Martin-Gay, Intermediate Algebra, 5ed 2424

Exponents with Rational Numbers

So far, we have only worked with integer exponents.

In this section, we extend exponents to rational numbers as a shorthand notation when using radicals.

The same rules for working with exponents will still apply.

Martin-Gay, Intermediate Algebra, 5ed 2525

Recall that a cube root is defined so that

abba 33 ifonly

However, if we let b = a1/3, then aaaab 133/133/13 )(

Since both values of b give us the same a,33/1 aa

Understanding a1/n

nn aa /1

n aIf n is a positive integer greater than 1 and is a real number, then

1Definition of na

Martin-Gay, Intermediate Algebra, 5ed 2626

Use radical notation to write the following. Simplify if possible.

3381 4 44 4/181

5/11032x 25 1055 10 2232 xxx

3/1716x 3233 633 743 7 2222216 xxxxxx

Example:

Martin-Gay, Intermediate Algebra, 5ed 2727

Understanding am/n

If m and n are positive integers greater than 1 with m/n in lowest terms, then

mnn mnm aaa /

1

Definition of na

n aas long as is a real number

Martin-Gay, Intermediate Algebra, 5ed 2828

Use radical notation to write the following. Simplify if possible.

3/48 4

3 8

3/773x 3 773x

32 73)73( xx

4

3 32 42 16

33 6 7373 xx

Example:

Martin-Gay, Intermediate Algebra, 5ed 2929

Understanding am/n

as long as a-m/n is a nonzero real number.

nmnm

aa

// 1

Definition of-m

na

Martin-Gay, Intermediate Algebra, 5ed 3030

Use radical notation to write the following. Simplify if possible.

3/264 3/264

1

4/516 4/516

1

23 64

1

23 34

1

24

116

1

54 42

1

52

1

32

1

Example:

Martin-Gay, Intermediate Algebra, 5ed 3131

Use properties of exponents to simplify the following. Write results with only positive exponents.

33/25/132 x 25/332 x

3/2

2/14/1

a

aa 3/22/14/1a

23

5 52 x 232 x 28x

12/812/612/3a 12/11a 12/11

1

a

Using Rules for Exponents

Example:

Martin-Gay, Intermediate Algebra, 5ed 3232

Use rational exponents to write as a single radical.

253 2/13/1 25 6/36/2 25 6/132 25 6 200

Using Rational Exponents

Example:

Martin-Gay, Intermediate Algebra, 5ed 3333

§ 7.3

Martin-Gay, Intermediate Algebra, 5ed 3434

n n na b ab

Product Rule for Radicals

Product Rule for Radicals

n a n bIf and are real numbers, then

Martin-Gay, Intermediate Algebra, 5ed 3535

Simplify the following radical expressions.

40 104 102

16

5 16

5

4

5

15 No perfect square factor, so the radical is already simplified.

Example:

Martin-Gay, Intermediate Algebra, 5ed 3636

Simplify the following radical expressions.

7x xx6 xx6 xx3

16

20

x

16

20

x

8

54

x 8

52

x

Example:

Martin-Gay, Intermediate Algebra, 5ed 3737

nn

n

a ab b

n a n bIf and are real numbers,

and is not zero, thenn b

Quotient Rule for Radicals

Martin-Gay, Intermediate Algebra, 5ed 3838

Simplify the following radical expressions.

3 16 3 28 33 28 3 2 2

3

64

3 3

3

64

3

4

33

Example:

Martin-Gay, Intermediate Algebra, 5ed 3939

The distance d between two points (x1,y1) and (x2,y2) is given by

212

212 yyxxd

The Distance Formula

Distance Formula

Martin-Gay, Intermediate Algebra, 5ed 4040

Find the distance between (5, 8) and (2, 2).

212

212 yyxxd

22 28)2(5 d

22 63 d

5345369 d

The Distance Formula

Example:

Martin-Gay, Intermediate Algebra, 5ed 4141

The midpoint of the line segment whose endpoints are (x1,y1) and (x2,y2) is the point with coordinates

1 2 1 2,2 2

x x y y

The Midpoint Formula

Midpoint Formula

Martin-Gay, Intermediate Algebra, 5ed 4242

Find the midpoint of the line segment that joins points P(5, 8) and P(2, 2).

1 2 1 2,2 2

x x y y

The Midpoint Formula

5 2( ) ( ) ,82

22

7 10,2 2

3.5,5

Example:

Martin-Gay, Intermediate Algebra, 5ed 4343

§ 7.4

Expressions

Martin-Gay, Intermediate Algebra, 5ed 4444

Sums and Differences

Rules in the previous section allowed us to split radicals that had a radicand which was a product or a quotient.

We can NOT split sums or differences.

baba

baba

Martin-Gay, Intermediate Algebra, 5ed 4545

In previous chapters, we’ve discussed the concept of “like” terms.

These are terms with the same variables raised to the same powers.

They can be combined through addition and subtraction.

Similarly, we can work with the concept of “like” radicals to combine radicals with the same radicand.Like radicals are radicals with the same index and the same radicand.

Like radicals can also be combined with addition or subtraction by using the distributive property.

Martin-Gay, Intermediate Algebra, 5ed 4646

373 38

24210 26

3 2 42 Can not simplify

35 Can not simplify

Example:

Martin-Gay, Intermediate Algebra, 5ed 4747

Simplify the following radical expression. 331275

3334325

3334325

333235

3325 36

Example:

Martin-Gay, Intermediate Algebra, 5ed 4848

Simplify the following radical expression.

91464 33

9144 3 3 145

Example:

Martin-Gay, Intermediate Algebra, 5ed 4949

Simplify the following radical expression. Assume that variables represent positive real numbers.

xxx 5453 3 xxxx 5593 2

xxxx 5593 2

xxxx 5533

xxxx 559

xxx 59 xx 510

Example:

Martin-Gay, Intermediate Algebra, 5ed 5050

nnn abba

0 if b b

a

b

an

n

n

n a n bIf and are real numbers,

Multiplying and Dividing Radical Expressions

Martin-Gay, Intermediate Algebra, 5ed 5151

Simplify the following radical expressions.

xy 53 xy15

23

67

ba

ba

23

67

ba

ba44ba 22ba

Multiplying and Dividing Radical Expressions

Example:

Martin-Gay, Intermediate Algebra, 5ed 5252

§ 7.5

Rationalizing Numerators and Denominators of Radical Expressions

Martin-Gay, Intermediate Algebra, 5ed 5353

Many times it is helpful to rewrite a radical quotient with the radical confined to ONLY the numerator.

If we rewrite the expression so that there is no radical in the denominator, it is called rationalizing the denominator.

This process involves multiplying the quotient by a form of 1 that will eliminate the radical in the denominator.

Rationalizing the Denominator

Martin-Gay, Intermediate Algebra, 5ed 5454

Rationalize the denominator.

2

3

2

2

3 9

6

3

3

3

3

22

23

2

6

33

3

39

3 6

3

3

27

3 6

3

3 6 33 3 2

Rationalizing the Denominator

Example:

Martin-Gay, Intermediate Algebra, 5ed 5555

Many rational quotients have a sum or difference of terms in a denominator, rather than a single radical.

In that case, we need to multiply by the conjugate of the numerator or denominator (which ever one we are rationalizing).

The conjugate uses the same terms, but the opposite operation (+ or ).

Conjugates

Martin-Gay, Intermediate Algebra, 5ed 5656

Rationalize the denominator.

32

23

332322

3222323

32

32

32

322236

1

322236

322236

Rationalizing the Denominator

Example:

Martin-Gay, Intermediate Algebra, 5ed 5757

An expression rewritten with no radical in the numerator is called rationalizing the numerator.

7

12

7

12

12

12

212

12

127

1212 84

12

214

12

214

12

21

6

Rationalizing the Numerator

Example:

Martin-Gay, Intermediate Algebra, 5ed 5858

Rationalize the numerator.

3

7

9y3

3

7

9y

3 2

3 2

3

3

y

y

3 23

3 23

37

39

y

yy

3 2

3 3

21

27

y

y3 221

3

y

y

yx

yx

yx

yx

yxyxyx

yxyxyx

yyxyxx

yyxyxx

yxyx

yx

2

Rationalizing the Numerator

Example:

Martin-Gay, Intermediate Algebra, 5ed 5959

§ 7.6

Radical Equations and Problem Solving

Martin-Gay, Intermediate Algebra, 5ed 6060

The Power Rule

Power RuleIf both sides of an equation are raised to the same power, solutions of the new equation contain all the solutions of the original equation, but might also contain additional solutions.

A proposed solution of the new equation that is NOT a solution of the original equation is an extraneous solution.

Martin-Gay, Intermediate Algebra, 5ed 6161

Solving a Radical Equation1) Isolate one radical on one side of the equation.

2) Raise each side of the equation to a power equal to the index of the radical and simplify.

3) If the equation still contains a radical term, repeat Steps 1 and 2. If not, solve the equation.

4) Check all proposed solutions in the original equation.

Martin-Gay, Intermediate Algebra, 5ed 6262

Solve the following radical equation.

011 x

11 x

2211 x

11x

2x

2 1 1 0 011

011 true

Substitute into the original equation.

So the solution is x = 2.

Example:

Martin-Gay, Intermediate Algebra, 5ed 6363

Solve the following radical equation.

812 xx

xx 281

22281 xx

2432641 xxx 2433630 xx )421)(3(0 xx

213 or 4

x

Example:

Martin-Gay, Intermediate Algebra, 5ed 6464

Substitute the value for x into the original equation, to check the solution.

3 32( ) 1 8 846 true

21 21 14

24

8

84

25

2

21

82

5

2

21

82

26 falseSo the solution is x = 3.

Example continued:

Martin-Gay, Intermediate Algebra, 5ed 6565

Solve the following radical equation.425 yy

22425 yy

44445 yyy

445 y

44

5 y

22

44

5

y

416

25y

16

89

16

254 y

Example:

Martin-Gay, Intermediate Algebra, 5ed 6666

Substitute the value for x into the original equation, to check the solution.

5 289 891

46 16

16

252

16

169

4

52

4

13

4

3

4

13 false So the solution is .

Example continued:

Martin-Gay, Intermediate Algebra, 5ed 6767

Solve the following radical equation.24342 xx

43242 xx

2243242 xx

43434442 xxx

4343842 xxx

43412 xx

22 43412 xx6448)43(16144242 xxxx

080242 xx

0420 xx

20or 4x

Example:

Martin-Gay, Intermediate Algebra, 5ed 6868

Substitute the value for x into the original equation, to check the solution.

2( ) 4 3( 4 24 4) 2164

242

true

2( ) 4 3( ) 420 20 2

26436

286

true

So the solution is x = 4 or 20.

Example continued:

Martin-Gay, Intermediate Algebra, 5ed 6969

Solve the following radical equation.

51 x

2251 x

251x

24x

24 1 5

525 true

Substitute into the original equation.

So the solution is x = 24.

Example:

Martin-Gay, Intermediate Algebra, 5ed 7070

Solve the following radical equation.

55 x

2255 x

255 x

5x

5 55 525

Does NOT check, since the left side of the equation is asking for the principal square root.

So the solution is .

Substitute into the original equation.

Example:

Martin-Gay, Intermediate Algebra, 5ed 7171

Pythagorean Theorem

If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then

a2 + b2 = c2

There are several applications in this section that require the use of the Pythagorean Theorem in order to solve.

The Pythagorean Theorem

Martin-Gay, Intermediate Algebra, 5ed 7272

Find the length of the hypotenuse of a right triangle when the length of the two legs are 2 inches and 7 inches.

c2 = 22 + 72 = 4 + 49 = 53

53c = inches

Using the Pythagorean Theorem

Example:

Martin-Gay, Intermediate Algebra, 5ed 7373

§ 7.7

Complex Numbers

Martin-Gay, Intermediate Algebra, 5ed 7474

Imaginary Numbers

Previously, when we encountered square roots of negative numbers in solving equations, we would say “no real solution” or “not a real number”.

Imaginary UnitThe imaginary unit i, is the number whose square is – 1. That is,

2 1 and 1i i

Martin-Gay, Intermediate Algebra, 5ed 7575

Write the following with the i notation.

25 125

32 132 1216

121 1121

5 i

11 i

24 i 24 i

The Imaginary Unit, i

Example:

Martin-Gay, Intermediate Algebra, 5ed 7676

Real numbers and imaginary numbers are both subsets of a new set of numbers.

Complex Numbers

A complex number is a number that can be written in the form a + bi, where a and b are real numbers.

Complex Numbers

Martin-Gay, Intermediate Algebra, 5ed 7777

Complex numbers can be written in the form a + bi (called standard form), with both a and b as real numbers.

a is a real number and bi would be an imaginary number.

If b = 0, a + bi is a real number.

If a = 0, a + bi is an imaginary number.

Standard Form of Complex Numbers

Martin-Gay, Intermediate Algebra, 5ed 7878

Write each of the following in the form of a complex number in standard form a + bi.

6 = 6 + 0i

8i = 0 + 8i

24 164 62 i

256 1256 6 + 5i

620 i

Standard Form of Complex Numbers

Example:

Martin-Gay, Intermediate Algebra, 5ed 7979

Sum or Difference of Complex Numbers

If a + bi and c + di are complex numbers, then their sum is

(a + bi) + (c + di) = (a + c) + (b + d)i

Their difference is

(a + bi) – (c + di) = (a – c) + (b – d)i

Adding and Subtracting Complex Numbers

Martin-Gay, Intermediate Algebra, 5ed 8080

Add or subtract the following complex numbers. Write the answer in standard form a + bi.

(4 + 3) + (6 – 2)i = 7 + 4i

(8 + 2i) – (4i) = (8 – 0) + (2 – 4)i = 8 – 2i

(4 + 6i) + (3 – 2i) =

Adding and Subtracting Complex Numbers

Example:

Martin-Gay, Intermediate Algebra, 5ed 8181

The technique for multiplying complex numbers varies depending on whether the numbers are written as single term (either the real or imaginary component is missing) or two terms.

Multiplying Complex Numbers

Martin-Gay, Intermediate Algebra, 5ed 8282

Note that the product rule for radicals does NOT apply for imaginary numbers.

2516 ii 54 220i )1(20 20

2516 2516 20400

Multiplying Complex Numbers

Martin-Gay, Intermediate Algebra, 5ed 8383

Multiply the following complex numbers.

8i · 7i

56i2

56(1)

56

Multiplying Complex Numbers

Example:

Martin-Gay, Intermediate Algebra, 5ed 8484

Multiply the following complex numbers. Write the answer in standard form a + bi.

5i(4 – 7i)

20i – 35i2

20i – 35(–1)

20i + 35

35 + 20i

Multiplying Complex Numbers

Example:

Martin-Gay, Intermediate Algebra, 5ed 8585

Multiply the following complex numbers. Write the answer in standard form a + bi.

(6 – 3i)(7 + 4i)

42 + 24i – 21i – 12i2

42 + 3i – 12(–1)

42 + 3i + 12

54 + 3i

Multiplying Complex Numbers

Example:

Martin-Gay, Intermediate Algebra, 5ed 8686

Complex Conjugates

The complex numbers (a + bi) and (a – bi) are complex conjugates of each other, and

(a + bi)(a – bi) = a2 + b2

Complex Conjugate

Martin-Gay, Intermediate Algebra, 5ed 8787

The conjugate of a + bi is a – bi.

The conjugate of a – bi is a + bi.

The product of (a + bi) and (a – bi) is

(a + bi)(a – bi)

a2 – abi + abi – b2i2

a2 – b2(–1)

a2 + b2, which is a real number.

Complex Conjugate

Martin-Gay, Intermediate Algebra, 5ed 8888

Use complex conjugates to divide the following complex numbers. Write the answer in standard form.

i

i

34

26

i

i

i

i

34

34

34

26

2

2

9121216

681824

iii

iii

)1(916

)1(62624 i

25

2618 ii

25

26

25

18

Dividing Complex Numbers

Example:

Martin-Gay, Intermediate Algebra, 5ed 8989

Divide the following complex numbers.

i6

5

i

i

i 6

6

6

5

236

30

i

i

)1(36

30i

36

30ii

6

5

Dividing Complex Numbers

Example:

Martin-Gay, Intermediate Algebra, 5ed 9090

1i

12 i

iiiii )1(23

1)1)(1(224 iii

iiiii )1(45

1)1)(1(246 iii

iiiii ))(1(347

1)1)(1(448 iii

The powers recycle through each multiple of 4.

14 ki

Patterns of i

Martin-Gay, Intermediate Algebra, 5ed 9191

Simplify each of the following powers.

53i ii52 ii 134 i1 i

17i 17

1

i

ii16

1

i1

1

i

1

i

i

i

1

2i

i

)1(

i

1

ii

Patterns of i

Example: