Rational Exponents, Radicals, and Complex Numbers Chapter 8 Rational Exponents, Radicals, and Complex Numbers 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8

Radicals and Radical Functions 8.1 Radicals and Radical Functions Objectives: Find square roots. Approximate roots. Find cube roots. Find nth roots. Find nth root of an. Graph square root and cube root functions.

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.

Example Simplify. Assume that all variables represent positive numbers. a. b. c. d. e. f. g. h.

 

Cube Root The cube root of a real number a is written as Nth Root The nth root of a real number a is written as

Example Find the roots. a. b. c. d. e. f. g. h.

If n is an even positive integer, then If n is an odd positive integer, then

Example Simplify. a. b. c. d. e. = 3x – 5 f. g.

Example If and find each function value. a. b. c. d. a. b. c. d.

Example Graph the square root function Domain: all nonnegative numbers, x ≥ 0. x y x y (6, ) (4, 2) (2, ) (1, 1) 1 (0, 0) 2 4 2 6

Example Graph the Domain: all real numbers x y 2 8 4 1 -1 -4 -2 -8 y (8, 2) (4, ) (1, 1) 1 (-1, -1) (0, 0) (-4, ) -1 (-8, -2) -4 -2 -8

Example Graph the square root function Domain: all real numbers. Vertex: (___, ___)   x y (0, 7) (4, 7) (3, 4) x y (1, 4) (2, 3) 7 1 4 2 3 vertex 3 4 4 7

Example Graph the square root function Domain: set what is under the radical > 0 and solve, x ≥ 4. Starting point (__, __)   x y x y 4 2 (8, 4) (5, 3) 5 3 (4, 2) 6 7 8 4

Example Graph the Domain: all real numbers x y -2 8 1 -3 -4 -1 -5 -6   x y x y -2 8 1 -3 -4 (8, -2) -1 -5 (1, -3) -6 -8 (-1, -5) (0, -4) (-8, -6)

Radicals and Radical Functions 8.1 Summary Radicals and Radical Functions Objectives: Find square roots. Approximate roots. Find cube roots. Find nth roots. Find nth root of an. Graph square root and cube root functions.

Rational Exponents 8.2 Objectives: Understand the meaning of a1/n, am/n and a-m/n Use rules for exponents to simplify expressions that contain rational exponents Use rational exponents to simplify radical expressions

Definition of a1/n If n is a positive integer greater than 1 and is a real number, then

Example Use radical notation to write the following. Simplify if possible. 811/4 (32x10)1/5 (16x7)1/3                     16x7 = 2∙2∙2∙2∙x∙x∙x∙x∙x∙x∙x

If m and n are positive integers greater than 1 with m/n in lowest terms, then as long as is a real number. as long as am/n is a nonzero real number.

Example Use radical notation to write the following. Then simplify if possible. a. b. c.       8

Example Write each expression with a positive exponent, and then simplify. a. b.

Example Use properties of exponents to simplify. Write results with only positive exponents. a. b. c.

Example Use properties of exponents to simplify. Write results with only positive exponents. a. b.    

Example Use rational exponents to simplify. Assume that variables represent positive numbers. a. b.

Example Use rational exponents to write as a single radical.

Rational Exponents 8.2 Summary Objectives: Understand the meaning of a1/n, am/n and a-m/n Use rules for exponents to simplify expressions that contain rational exponents Use rational exponents to simplify radical expressions Do now: multiples of 4 #4-20

Simplifying Radical Expressions 8.3 Simplifying Radical Expressions Objectives: Use the product and quotient rule for radicals Simplify radicals Use the distance and midpoint formulas

Product Rule for Radicals If are real numbers, then Quotient Rule for Radicals If are real numbers and is not zero, then

Example Multiply. a. b. c. d.                

Example Multiply. a. Multiply. a. b. c. d.

Example Simplify the following. a. b. c. No perfect square factor, so the radical is already simplified. 1 1 5 5 3 3 2 10 3 9 2 20 3 27 2 40 5 135

Example Use the product rule to simplify. a. b.    

Example Use the quotient rule to simplify. a. b.   x2    

Distance Formula The distance d between two points (x1, y1) and (x2, y2) is given by Midpoint Formula The midpoint of the line segment whose endpoints are (x1, y1) and (x2, y2) is the point with coordinates

Example Find the distance between (2, 2) and (5, 8). There way My way   Dist between y’s 6   Dist between x’s 3 Then use Pythagorean Thm.

Example Find the midpoint between (2, 2) and (5, 8).

Find the distance and midpoint

Simplifying Radical Expressions 8.3 Summary Simplifying Radical Expressions Objectives: Use the product and quotient rule for radicals Simplify radicals Use the distance and midpoint formulas

Adding, Subtracting, and Multiplying Radical Expressions 8.4 Adding, Subtracting, and Multiplying Radical Expressions Objectives: Add/Subtract radical expressions Multiply radical expressions

Radicals with the same index and the same radicand are like radicals. Unlike radicals

Example Add or subtract as indicated. Assume all variables represent positive real numbers. a. b. c. d. Cannot simplify Cannot simplify

Example Add or subtract. Assume that variables represent positive real numbers. a. b.    

Example Add or subtract. Assume that variables represent positive real numbers.  

Example Simplify. Assume that variables represent positive real numbers. =

Example Multiply. a. b.          

Example Multiply.                

Example Multiply.        

Adding, Subtracting, and Multiplying Radical Expressions 8.4 summary Adding, Subtracting, and Multiplying Radical Expressions Objectives: Add/Subtract radical expressions Multiply radical expressions

Rationalizing Denominators and Numerators of Radical Expressions 8.5 Rationalizing Denominators and Numerators of Radical Expressions Objectives: Rationalize denominators with one or two terms Rationalize numerators

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 that will eliminate the radical in the denominator.

Example Rationalize the denominator of each expression. a. b.

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 ).

Example Rationalize the denominator.        

Example Rationalize the denominator. +2

Sometimes a numerator has to be rationalized An expression rewritten with no radical in the numerator is called rationalizing the numerator.

Example Rationalize the numerator of      

Example Rationalize the numerator of                

Rationalizing Denominators and Numerators of Radical Expressions 8.5 summary Rationalizing Denominators and Numerators of Radical Expressions Objectives: Rationalize denominators with one or two terms Rationalize numerators

Radical Equations and Problem Solving 8.6 Radical Equations and Problem Solving Objectives: Solve equations that contain radical expressions Use the Pythagorean Theorem to model problems

Power Rule If both sides of an equation are raised to the same power, all solutions of the original equation are among the solutions of the new equation. A proposed solution of the new equation that is NOT a solution of the original equation is an extraneous solution.

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. 3. Check all proposed solutions in the original equation.

Example Solve: Check: The solution is 13 or the solution set is {13}.

Example Solve: Check: True The solution is x = 2.

Example Solve: true     false The solution is x = 3.

Reminder how to do the following: Multiply.        

Example Solve: false The solution is . 25 = 16(y – 4) 25 = 16y – 64   false 25 = 16(y – 4) 25 = 16y – 64 The solution is . 89 = 16y 89/16 = y

Solve: The solutions are x = 4 or 20. Example true –3x – 8 –4 –3x – 4   true   The solutions are x = 4 or 20.

Example Solve: Check: Substitute into the original equation. false The solution is .

Example       And check the solution:  

Example Find the length of the unknown leg of the right triangle. b 14 m Find the length of the unknown leg of the right triangle. The unknown leg of the triangle is

Example A 50-foot supporting wire is to be attached to a 75-foot antenna. Because of surrounding buildings, sidewalks, and roadways, the wire must be anchored exactly 20 feet from the base of the antenna. a. How high from the base of the antenna is the wire attached? Local regulations require that a supporting wire be attached at a height no less than 3/5 of the total height of the antenna. Have local regulations been met? The wire is attached ≈ 45.8 ft from the base of the pole. The supporting wire must be attached at a height no less than 3/5(75 feet), or 45 feet. Since we know from part a that the wire is to be installed at a height of approximately 45.8 feet, local regulations have been met.

Radical Equations and Problem Solving 8.6 summary Radical Equations and Problem Solving Objectives: Solve equations that contain radical expressions Use the Pythagorean Theorem to model problems

Complex Numbers 8.7 Objectives: Write square roots of negative numbers in the form bi Graph complex numbers on the complex plane Add, subtract, multiply, and divide complex numbers Raise i to powers

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, written i, is the number whose square is – 1. That is,

Example Write with i notation. a. b. c.

Example Multiply or divide as indicated. a. b. c.

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

Recall that the graph of a real number is a point on a real number line. In the same manner, the graph of a complex number is a point in the complex plane. The horizontal axis is the real axis. The vertical axis is the imaginary axis.

Example Graph the complex numbers in the complex plane. a. 2 + 3i b. 5 + 4i c. 3i d. 5

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

Example Add or subtract the complex numbers. Write the sum or difference in the form a + bi. a. (4 + 6i) + (3 – 2i) = b. (8 + 2i) – (4i) = (4 + 3) + (6 – 2)i = 7 + 4i (8 – 0) + (2 – 4)i = 8 – 2i

Example Multiply the complex numbers. Write the product in the form a + bi. a. 4i  6i b. c. 5i(8 – 4i) = 5i(8) – 5i(4i) = 40i – 20i2 = 40i – 20(–1) = 40i + 20 (6 – i)(2 + i) F O I L = 12 + 6i – 2i – i2 = 12 + 4i – (–1) = 12 + 4i + 1 = 13 + 4i = –24i2 = –24(–1) =24

Complex Conjugates The complex numbers (a + bi) and (a – bi) are called complex conjugates of each other, and (a + bi)(a – bi) = a2 + b2.

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.

Example Divide. Write in the form a + bi. i = 5i 6i2

Patterns of i

Example Find the following powers of i.

Complex Numbers 8.7 summary Do the following now: #4, 8, 12, 16, 20, 24 Objectives: Write square roots of negative numbers in the form bi Graph complex numbers on the complex plane Add, subtract, multiply, and divide complex numbers Raise i to powers

Standard Deviation 8.8 Objectives: Review mean, median, mode, and range Determine the standard deviation for a data set

Recall that it is sometimes desirable to be able to describe a set of data, or a set of numbers, by a single “middle” number. Three such measures of central tendency are the mean, the median, and the mode. The mean is the most common measure of central tendency.

Measures of dispersion are used to describe the spread of data items in a data set. Two of the most common measures of dispersion are the range and the standard deviation.

The range, the difference between the highest and the lowest The mean is the sum of the data items divided by the number of items Mean = = sum of data values n = number of data values The median of an ordered set of numbers is the middle number. If the number of items is even, the median is the mean of the two middle numbers. The mode of a set of numbers is the number that occurs most often. It is possible for a data set to have no mode or more than one mode. The range, the difference between the highest and the lowest data values in a data set, indicates the total spread of the data. Range = highest data value – lowest data value

Example Seven students in a class conducted an experiment on “how long is a minute”. Each student was asked to close their eyes and tell their partner when they thought one minute was up. Their actual time was recorded below. a. What was the longest time? b. What was the shortest time? c. Find the mean. d. How many students took longer than the mean time? How many took shorter than the mean time? Jessie at 83 seconds Carmen at 36 seconds Student Anna Brad Carmen David Ethan Frankie Jessie Time (seconds) 45 72 36 62 56 49 83

Four students were shorter than the mean time Four students were shorter than the mean time. Three students were longer than the mean time. Student Anna Brad Carmen David Ethan Frankie Jessie Time (seconds) 45 72 36 62 56 49 83

Example Find the median and mode of the following list of numbers. 66, 70, 75, 76, 79, 77, 72, 67, 66, 69, 72, 79, 79, 82 Write the numbers in order. 66, 66, 67, 69, 70, 72, 72, 75, 76, 77, 79, 79, 79, 82 two middle numbers mode The mode is 79, since 79 occurs most often.

Example Honolulu’s hottest day is 89º and its coldest day is 61º. What is the range in temperatures? 89º  61º = 28º The range is 28º.

Computing the Standard Deviation for a Data Set Find the mean of the data items. Find the deviation of each data item from the mean: data item – mean. Square each deviation: (data item – mean)2. Sum the squared deviations: Σ(data item – mean)2. Divide the sum in step 4 by n  1, where n represents the number of data items: Take the square root of the quotient in step 5. This value is the standard deviation for the data set. Standard deviation =

Deviation = data item – mean The graph shows the number of workers, in millions, for the five countries with the largest labor forces. Find the standard deviation. Data Deviation = data item – mean (Deviation)² (data item – mean)2 778 778 – 317 = 461 461² = 212,521 472 472 – 317 = 155 155² = 24,025 147 147 – 317 = –170 (–170)² = 28,900 106 106 – 317 = –211 (–211)² = 44,521 82 82 – 317 = –235 (– 235)²= 55,225 6/3/2018 Section 12.3

Standard Deviation 8.8 Summary Objectives: Review mean, median, mode, and range Determine the standard deviation for a data set