Chapter 7 Rational Exponents Radicals and Complex Numbers

Chapter 7 Rational Exponents, Radicals, and Complex Numbers

§ 7. 1 Radicals and Radical Functions

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. In order to find a square root of a, you need a # that, when squared, equals a. Martin-Gay, Intermediate Algebra, 5 ed 3

Principal Square Roots Principal and Negative Square Roots If a is a nonnegative number, then is the principal or nonnegative square root of a is the negative square root of a. Martin-Gay, Intermediate Algebra, 5 ed 4

Radicands 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, 5 ed 5

Radicands Example: Martin-Gay, Intermediate Algebra, 5 ed 6

Perfect Squares 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. Martin-Gay, Intermediate Algebra, 5 ed 7

Perfect Square Roots 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. Example: Martin-Gay, Intermediate Algebra, 5 ed 8

Cube Roots Cube Root The cube root of a real number a is written as Martin-Gay, Intermediate Algebra, 5 ed 9

Cube Roots Example: Martin-Gay, Intermediate Algebra, 5 ed 10

nth Roots Other roots can be found, as well. The nth root of a is defined as 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. Martin-Gay, Intermediate Algebra, 5 ed 11

nth Roots Example: Simplify the following. Martin-Gay, Intermediate Algebra, 5 ed 12

nth Roots Example: Simplify the following. Assume that all variables represent positive numbers. Martin-Gay, Intermediate Algebra, 5 ed 13

nth Roots If the index of the root is even, then the notation represents a positive number. 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. Martin-Gay, Intermediate Algebra, 5 ed 14

Finding nth Roots If n is an even positive integer, then If n is an odd positive integer, then Martin-Gay, Intermediate Algebra, 5 ed 15

Finding nth Roots Simplify the following. If we know for sure that the variables represent positive numbers, we can write our result without the absolute value sign. Martin-Gay, Intermediate Algebra, 5 ed 16

Finding nth Roots Example: 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. Martin-Gay, Intermediate Algebra, 5 ed 17

Evaluating Rational Functions 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. Example: Find the value Martin-Gay, Intermediate Algebra, 5 ed 18

Root Functions Since every value of x that is substituted into the equation 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. Martin-Gay, Intermediate Algebra, 5 ed 19

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. Martin-Gay, Intermediate Algebra, 5 ed 20

Graphs of Root Functions Example: y Graph x 6 4 2 1 0 y 2 (2, (1, 1) (0, 0) ) (4, 2) (6, ) x 1 0 Martin-Gay, Intermediate Algebra, 5 ed 21

Graphs of Root Functions Example: Graph x y 8 4 2 1 1 0 -1 -4 -8 0 -1 y (1, 1) (-1, -1) (-8, -2) (-4, (0, 0) ) (8, 2) x ) -2 Martin-Gay, Intermediate Algebra, 5 ed 22

§ 7. 2 Rational Exponents Martin-Gay, Intermediate Algebra, 5 ed 23

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, 5 ed 24

Understanding 1/n a Recall that a cube root is defined so that However, if we let b = a 1/3, then Since both values of b give us the same a, If n is a positive integer greater than 1 and real number, then Martin-Gay, Intermediate Algebra, 5 ed is a 25

Using Radical Notation Example: Use radical notation to write the following. Simplify if possible. Martin-Gay, Intermediate Algebra, 5 ed 26

Understanding m/n a If m and n are positive integers greater than 1 with m/n in lowest terms, then as long as is a real number Martin-Gay, Intermediate Algebra, 5 ed 27

Using Radical Notation Example: Use radical notation to write the following. Simplify if possible. Martin-Gay, Intermediate Algebra, 5 ed 28

Understanding m/n a as long as a-m/n is a nonzero real number. Martin-Gay, Intermediate Algebra, 5 ed 29

Using Radical Notation Example: Use radical notation to write the following. Simplify if possible. Martin-Gay, Intermediate Algebra, 5 ed 30

Using Rules for Exponents Example: Use properties of exponents to simplify the following. Write results with only positive exponents. Martin-Gay, Intermediate Algebra, 5 ed 31

Using Rational Exponents Example: Use rational exponents to write as a single radical. Martin-Gay, Intermediate Algebra, 5 ed 32

§ 7. 3 Simplifying Radical Expressions Martin-Gay, Intermediate Algebra, 5 ed 33

Product Rule for Radicals If and are real numbers, then Martin-Gay, Intermediate Algebra, 5 ed 34

Simplifying Radicals Example: Simplify the following radical expressions. No perfect square factor, so the radical is already simplified. Martin-Gay, Intermediate Algebra, 5 ed 35

Simplifying Radicals Example: Simplify the following radical expressions. Martin-Gay, Intermediate Algebra, 5 ed 36

Quotient Rule Radicals Quotient Rule for Radicals If and are real numbers, Martin-Gay, Intermediate Algebra, 5 ed 37

Simplifying Radicals Example: Simplify the following radical expressions. Martin-Gay, Intermediate Algebra, 5 ed 38

The Distance Formula The distance d between two points (x 1, y 1) and (x 2, y 2) is given by Martin-Gay, Intermediate Algebra, 5 ed 39

The Distance Formula Example: Find the distance between ( 5, 8) and ( 2, 2). Martin-Gay, Intermediate Algebra, 5 ed 40

The Midpoint Formula The midpoint of the line segment whose endpoints are (x 1, y 1) and (x 2, y 2) is the point with coordinates Martin-Gay, Intermediate Algebra, 5 ed 41

The Midpoint Formula Example: Find the midpoint of the line segment that joins points P( 5, 8) and P( 2, 2). Martin-Gay, Intermediate Algebra, 5 ed 42

§ 7. 4 Adding, Subtracting, and Multiplying Radical Expressions Martin-Gay, Intermediate Algebra, 5 ed 43

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. Martin-Gay, Intermediate Algebra, 5 ed 44

Like Radicals 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, 5 ed 45

Adding and Subtracting Radical Expressions Example: Can not simplify Martin-Gay, Intermediate Algebra, 5 ed 46

Adding and Subtracting Radical Expressions Example: Simplify the following radical expression. Martin-Gay, Intermediate Algebra, 5 ed 47

Adding and Subtracting Radical Expressions Example: Simplify the following radical expression. Martin-Gay, Intermediate Algebra, 5 ed 48

Adding and Subtracting Radical Expressions Example: Simplify the following radical expression. Assume that variables represent positive real numbers. Martin-Gay, Intermediate Algebra, 5 ed 49

Multiplying and Dividing Radical Expressions If and are real numbers, Martin-Gay, Intermediate Algebra, 5 ed 50

Multiplying and Dividing Radical Expressions Example: Simplify the following radical expressions. Martin-Gay, Intermediate Algebra, 5 ed 51

§ 7. 5 Rationalizing Numerators and Denominators of Radical Expressions Martin-Gay, Intermediate Algebra, 5 ed 52

Rationalizing the Denominator 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. Martin-Gay, Intermediate Algebra, 5 ed 53

Rationalizing the Denominator Example: Rationalize the denominator. Martin-Gay, Intermediate Algebra, 5 ed 54

Conjugates 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 ). Martin-Gay, Intermediate Algebra, 5 ed 55

Rationalizing the Denominator Example: Rationalize the denominator. Martin-Gay, Intermediate Algebra, 5 ed 56

Rationalizing the Numerator An expression rewritten with no radical in the numerator is called rationalizing the numerator. Example: Martin-Gay, Intermediate Algebra, 5 ed 57

Rationalizing the Numerator Example: Rationalize the numerator. Martin-Gay, Intermediate Algebra, 5 ed 58

§ 7. 6 Radical Equations and Problem Solving Martin-Gay, Intermediate Algebra, 5 ed 59

The Power Rule If 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, 5 ed 60

Solving Radical Equations Solving a Radical Equation 1) 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, 5 ed 61

Solving Radical Equations Example: Solve the following radical equation. Substitute into the original equation. true So the solution is x = 2. Martin-Gay, Intermediate Algebra, 5 ed 62

Solving Radical Equations Example: Solve the following radical equation. Martin-Gay, Intermediate Algebra, 5 ed 63

Solving Radical Equations Example continued: Substitute the value for x into the original equation, to check the solution. true So the solution is x = 3. Martin-Gay, Intermediate Algebra, 5 ed false 64

Solving Radical Equations Example: Solve the following radical equation. Martin-Gay, Intermediate Algebra, 5 ed 65

Solving Radical Equations Example continued: Substitute the value for x into the original equation, to check the solution. false So the solution is . Martin-Gay, Intermediate Algebra, 5 ed 66

Solving Radical Equations Example: Solve the following radical equation. Martin-Gay, Intermediate Algebra, 5 ed 67

Solving Radical Equations Example continued: Substitute the value for x into the original equation, to check the solution. true So the solution is x = 4 or 20. Martin-Gay, Intermediate Algebra, 5 ed 68

Solving Radical Equations Example: Solve the following radical equation. Substitute into the original equation. true So the solution is x = 24. Martin-Gay, Intermediate Algebra, 5 ed 69

Solving Radical Equations Example: Solve the following radical equation. Substitute into the original equation. Does NOT check, since the left side of the equation is asking for the principal square root. So the solution is . Martin-Gay, Intermediate Algebra, 5 ed 70

The 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 a 2 + b 2 = c 2 There are several applications in this section that require the use of the Pythagorean Theorem in order to solve. Martin-Gay, Intermediate Algebra, 5 ed 71

Using the Pythagorean Theorem Example: Find the length of the hypotenuse of a right triangle when the length of the two legs are 2 inches and 7 inches. c 2 = 22 + 72 = 4 + 49 = 53 c= inches Martin-Gay, Intermediate Algebra, 5 ed 72

§ 7. 7 Complex Numbers Martin-Gay, Intermediate Algebra, 5 ed 73

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 Unit The imaginary unit i, is the number whose square is – 1. That is, Martin-Gay, Intermediate Algebra, 5 ed 74

The Imaginary Unit, i Example: Write the following with the i notation. i i Martin-Gay, Intermediate Algebra, 5 ed 75

Complex Numbers 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. Martin-Gay, Intermediate Algebra, 5 ed 76

Standard Form of 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. Martin-Gay, Intermediate Algebra, 5 ed 77

Standard Form of Complex Numbers Example: Write each of the following in the form of a complex number in standard form a + bi. 6 = 6 + 0 i 8 i = 0 + 8 i i i 6 + 5 i Martin-Gay, Intermediate Algebra, 5 ed 78

Adding and Subtracting Complex Numbers 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 Martin-Gay, Intermediate Algebra, 5 ed 79

Adding and Subtracting Complex Numbers Example: Add or subtract the following complex numbers. Write the answer in standard form a + bi. (4 + 6 i) + (3 – 2 i) = (4 + 3) + (6 – 2)i = 7 + 4 i (8 + 2 i) – (4 i) = (8 – 0) + (2 – 4)i = 8 – 2 i Martin-Gay, Intermediate Algebra, 5 ed 80

Multiplying Complex Numbers 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. Martin-Gay, Intermediate Algebra, 5 ed 81

Multiplying Complex Numbers Note that the product rule for radicals does NOT apply for imaginary numbers. Martin-Gay, Intermediate Algebra, 5 ed 82

Multiplying Complex Numbers Example: Multiply the following complex numbers. 8 i · 7 i 56 i 2 56( 1) 56 Martin-Gay, Intermediate Algebra, 5 ed 83

Multiplying Complex Numbers Example: Multiply the following complex numbers. Write the answer in standard form a + bi. 5 i(4 – 7 i) 20 i – 35 i 2 20 i – 35(– 1) 20 i + 35 35 + 20 i Martin-Gay, Intermediate Algebra, 5 ed 84

Multiplying Complex Numbers Example: Multiply the following complex numbers. Write the answer in standard form a + bi. (6 – 3 i)(7 + 4 i) 42 + 24 i – 21 i – 12 i 2 42 + 3 i – 12(– 1) 42 + 3 i + 12 54 + 3 i Martin-Gay, Intermediate Algebra, 5 ed 85

Complex Conjugates The complex numbers (a + bi) and (a – bi) are complex conjugates of each other, and (a + bi)(a – bi) = a 2 + b 2 Martin-Gay, Intermediate Algebra, 5 ed 86

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) a 2 – abi + abi – b 2 i 2 a 2 – b 2(– 1) a 2 + b 2, which is a real number. Martin-Gay, Intermediate Algebra, 5 ed 87

Dividing Complex Numbers Example: Use complex conjugates to divide the following complex numbers. Write the answer in standard form. Martin-Gay, Intermediate Algebra, 5 ed 88

Dividing Complex Numbers Example: Divide the following complex numbers. Martin-Gay, Intermediate Algebra, 5 ed 89

Patterns of i The powers recycle through each multiple of 4. Martin-Gay, Intermediate Algebra, 5 ed 90

Patterns of i Example: Simplify each of the following powers. Martin-Gay, Intermediate Algebra, 5 ed 91