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Chapter 7 7.1 Radical Expression Theorem 7.1 • Every positive real number has two real number square roots • 0 has just one square root: 0 • Negative numbers do not have real number square roots Principal Square Root • The Principal Square Root of a non-negative real number is its non-negative root. • Thesymbol a represents the principal squareroot of a. • The negative square root of a is written as; a • This symbol is a radical sign • An expression written with a radical sign is a radical expression • The expression written under the radical sign is the radicand Radical Expression Index Radical k a Radicand Absolute Value • For any real number a, a a 2 • The principal square root of a2 is the absolute value of a. k a a ; When k is even k a a; When k is odd k k HW 7.1 Pg 295-296 1-39 Odd, 40-53 Chapter 7 7.2 Multiplying and Simplifying HW 7.2 Pg 299 1-47 Chapter 7 7.3 Operations with Radical Expressions Chapter 7 7.4 More Operations with Radical Expressions HW 7.3-4 Pg 303 1-52 Pg 308-309 1-65 Odd, 66-69 Chapter 7 7.5 Rational Numbers as Exponents Theorem 7.6: For any nonnegative number a, any natural number index k, and any integer m, k 3 3 a m a k m 27 2 ? 2 8 ? 2 3 6 6y ? 3 ? Definition: For any nonnegative number a and any natural 1 k number index k, a means k a (the nonnegative kth root of a) Note: When working with rational exponents, we will assume that variables in the base are nonnegative 1 2 x x 1 3 27 3 27 3 Definition: For any natural numbers m and k, and any m k nonnegative number a, a means 2 3 k 27 27 3 2 am 3 27 2 m m Definition: For any rational number k and any real number a, a k means 1 a m k (5 xy ) 4 5 1 (5 xy ) 4 5 HW #7.5 Pg 315-316 3-75 Every third problem, 78-83 Chapter 7 7.6 Solving Radical Equations Isolate the radical expression on one side of the equation. Note: When solving a radical equation it is sometimes necessary to isolate the radical and use the inverse operation more than once 2x 5 1 x 3 HW #7.6 Pg 319-320 3-39 every third problem 41-52 Chapter 7 7.7 Imaginary and Complex Numbers Solve: x2 + 1 = 0 x2 = -1 x 1 In the real number system negative numbers do not have square roots, therefore there is no real solution. Mathematicians invented imaginary numbers so negative numbers would have square roots Thus the solution to the above equation would be x i Definition: The imaginary numbers consist of all numbers bi, where b is a real number and i is the imaginary unit, with the property that i2 = -1 i 1 i 1 i 63 i 28 2 i i 3 i 53 i 1 4 i121 Pattern Recognition Using the information from above, write a general statement about the standard form of in where n is a positive integer. Use this statement to write i231in standard form. When operating on imaginary numbers: 1. Always take the i out of the radical first 2. Treat i as a variable 3. Never write i with a power greater than 1 1. 47i 2 94i 3. 6i 2i 12 2. 5 2i 2 5 4. 3 6 3 2 Definition: The complex numbers consist of all sums a + bi, where a and b are real numbers and i is the imaginary unit. The real part is a and the imaginary part is bi. All real numbers are complex numbers We assume that i behaves like a real number, that is it obeys all the rules of real numbers 34. 3(2 5i)2 (2 i)(4) 5i 55 69i HW #7.7 Pg 323 1-33 Odd 34-63 Chapter 7 7-8 Complex Numbers and Graphing Remember a complex number has a real part and an imaginary part. These are used to plot complex numbers on a complex plane. z a bi Imaginary Axis absolute value of z z a bi The denoted by |z| is the z a b Real Axis distance from the origin to the point (x, y). z a b 2 2 Plot the number in the complex plane and then find the absolut value of the complex number. Plot the number in the complex plane and then find the absolute value of the complex number. Adding and Subtracting Complex numbers Graphically Adding and Subtracting Complex numbers Graphically HW #7.8 Pg 325 1-23 Chapter 7 7-9 More about Complex Numbers Equality of Complex Numbers: a +bi = c + di if and only if a = c and b = d Solve the following equation for x and y: 5x + 6i = 10 + 2yi Equality of Complex Numbers: a +bi = c + di if and only if a = c and b = d Solve the following equations for x and y: 3x + yi = 5x + 1 + 2i Multiply Multiply Multiply Conjugate of a complex number The conjugate of a + bi is a - bi Theorem 7-8 The product of a nonzero complex number a + bi and its conjugate a - bi is a2 + b2 Prove Theorem 7-8 Find the reciprocal of 3 + 4i. HW #7.9 Pg 329 1-27 Odd, 28-41 Chapter 7 7-10 Solutions of Equations A Solution to an Equation is the number that when substituted for the variable makes the equations true. Is 2 + i a solution of x2 – 4x + 5 = 0? A Solution to an Equation is the number that when substituted for the variable makes the equations true. Is 1 i 7 a solution of x2 – 2x + 8 = 0? Principle of zero products: If ab = 0 , then a = 0 or b = 0. Find an equation having 1 + 2i and 1 – 2i as solutions Principle of zero products: If ab = 0 , then a = 0 or b = 0. Find an equation having 4 +3i and 4 – 3i as solutions Solve the following equations: 1. 3ix + 4 – 5i = (1 + i)x + 2i 2. 3 – 4i + 2ix = 3i - (1 – i)x 3. 3x + 4ix = 1 + i Theorem 7-9 Every polynomial with complex coefficients and a degree of n (where n >1) can be factored into n linear factors. Show that (x + i)(x – i) is a factorization of x2 + 1 Show that (x + 2i)(x – 2i) is a factorization of x2 + 4 Show that [x - (2 + 3i)][(x – (2 - 3i)] is a factorization of x2 – 4x + 13 Theorem 7-10 Every polynomial of degree n (n 1) with complex coefficients has at least one solution and at most n solutions in the complex number system. Theorem 7-11 Every nonzero complex number has two square roots. They are additive inverses of each other. 0 has one square root. Show that 1 + i is the square root of 2i. Find the other square root. Show that -1 + i is the square root of -2i. Find the other square root. Show that 3 - i is the square root of 8 - 6i. Find the other square root. HW #7.10 Pg 333 1-31 Odd, 32