Rational Exponents, Radicals, and Complex Numbers

Chapter 7

Rational Exponents, Radicals, and

Complex Numbers

§ 7.1

Radicals and Radical Functions

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.


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


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.

Radicands


49 7

16

25

4

5

4 2

Radicands

Example:


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


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:


Cube Root

The cube root of a real number a is written as

3 , anda

Cube Roots

abba 33 ifonly


3 27 3

3 68x 22x

Cube Roots

Example:


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


Simplify the following.

20225 ba 105ab

39

364

b

a3

4

b

a

nth Roots

Example:


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:


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


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



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


39

364

b

a3

4

b

a


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:


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


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


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


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:


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:


§ 7.2

Rational Exponents


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.


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


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

Using Radical Notation

Example:


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



3/48 4

3 8

3/773x 3 773x

32 73)73( xx

4

3 32 42 16

33 6 7373 xx


Example:


Understanding am/n

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

nmnm

aa

// 1

Definition of-m

na



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:


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:


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:


§ 7.3

Simplifying Radical Expressions


n n na b ab

Product Rule for Radicals

Product Rule for Radicals

n a n bIf and are real numbers, then


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.

Simplifying Radicals

Example:



7x xx6 xx6 xx3

16

20

x

16

20

x

8

54

x 8

52

x


Example:


nn

n

a ab b

n a n bIf and are real numbers,

Quotient Rule Radicals

and is not zero, thenn b

Quotient Rule for Radicals



3 16 3 28 33 28 3 2 2

3

64

3 3

3

64

3

4

33


Example:


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

212

212 yyxxd

The Distance Formula

Distance Formula


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:


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


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:


§ 7.4

Adding, Subtracting, and Multiplying Radical

Expressions


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


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.

Like Radicals


373 38

24210 26

3 2 42 Can not simplify

35 Can not simplify

Adding and Subtracting Radical Expressions

Example:


Simplify the following radical expression. 331275

3334325

3334325

333235

3325 36


Example:


Simplify the following radical expression.

91464 33

9144 3 3 145


Example:


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:


nnn abba

0 if b b

a

b

an

n

n

n a n bIf and are real numbers,

Multiplying and Dividing Radical Expressions



xy 53 xy15

23

67

ba

ba

23

67

ba

ba44ba 22ba

Multiplying and Dividing Radical Expressions

Example:


§ 7.5

Rationalizing Numerators and Denominators of Radical Expressions


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


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


Example:


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


Rationalize the denominator.

32

23

332322

3222323

32

32

32

322236

1

322236

322236


Example:


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:


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:


§ 7.6

Radical Equations and Problem Solving


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.


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.

Solving Radical Equations


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:



812 xx

xx 281

22281 xx

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

213 or 4

x


Example:


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:



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:



5 289 891

46 16

16

252

16

169

4

52

4

13

4

3

4

13 false So the solution is .


Example continued:


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:



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:



51 x

2251 x

251x

24x

24 1 5

525 true


So the solution is x = 24.


Example:



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 .



Example:


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


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:


§ 7.7

Complex Numbers


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


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:


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


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


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:


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


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:


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


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

2516 ii 54 220i )1(20 20

2516 2516 20400



Multiply the following complex numbers.

8i · 7i

56i2

56(1)

56


Example:


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


Example:


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


Example:


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


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


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:


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:


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


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:

Rational Exponents, Radicals, and Complex Numbers

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