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Section 7.1 Rational Exponents and Radicals OBJECTIVES A Find the nth root of a number, if it exists. OBJECTIVES B Evaluate expressions containing rational exponents. OBJECTIVES C Simplify expressions involving rational exponents. DEFINITION NTH ROOT If a and x are real numbers and n is a positive integer: x is an nth root of a if xn = a DEFINITION PRINCIPLE NTH ROOT If n is a positive integer, then na denotes the principle nth root of a DEFINITION RATIONAL EXPONENTS AND THEIR ROOTS If n is a positive integer and n a is a real number: a1/ n = n a DEFINITION RADICAL EXPRESSION WITH AN M/N EXPONENT m/ n n m n m a =( a) = a Provided m and n are positive integers and n a is a real number. LAWS OF EXPONENTS If r, s, and t are rational, and a and b are real: I. ar as = ar+s r a r –s II. s = a a LAWS OF EXPONENTS If r, s, and t are rational, and a and b are real: III. (ar )s = ar r s t s rt st IV. (a b ) = a b Chapter 7 Section 7.1A Practice Test Exercise #1 Find, if possible. 3 a. = 3 –64 – 4 = –4 3 Find, if possible. b. –36 It is not a real number. Chapter 7 Section 7.1B Practice Test Exercise #4 Evaluate if possible. a. –27 – 2 3 = = = 1 2 – 27 3 1 3 27 – 1 –3 2 = 2 1 9 Evaluate, if possible. b. 8 –2 3 1 = 8 = 2 3 1 3 1 = 22 2 8 = 1 4 Section 7.2 Simplifying Radicals OBJECTIVES A Simplify radical expressions. OBJECTIVES B Rationalize the denominator of a fraction. OBJECTIVES C Reduce the order of a radical expression. DEFINITION nTH ROOT 1/n a = n a (a 0) when n is a positive integer. LAWS For Simplifying Radical Expressions I. n n a =a (a 0) LAWS For Simplifying Radical Expressions II. n n n ab = a b Product rule (a, b 0) III. LAWS For Simplifying Radical Expressions n a b n = n a b Quotient rule (a 0 ,b > 0) DEFINITION n an n n a = |a | n is an even positive integer DEFINITION n an n n a =a n is an odd positive integer Chapter 7 Section 7.2A Practice Test Exercise #7b Simplify. 4 – x 4 b. = 4 –x = –x = x 4 Chapter 7 Section 7.2A Practice Test Exercise #8b Simplify. 3 b. = = 54a 4 b12 3 3 27 • 2 • a3 • a • b12 33 a3 b12 • 2a = 3ab 4 3 2a Chapter 7 Section 7.2A Practice Test Exercise #9b Simplify. 3 b. 5 x6 3 = 3 3 = 5 x6 5 x2 Section 7.3 Operations with Radicals OBJECTIVES A Add and subtract similar radical expressions. OBJECTIVES B Multiply and divide radical expressions. OBJECTIVES C Rationalize the denominators of radical expressions involving sums or differences. DEFINITION LIKE RADICAL EXPRESSIONS Radical expressions with the same index and the same radicand. DEFINITION CONJUGATE The expressions a + b and a – b are conjugates of each other. Chapter 7 Section 7.3A Practice Test Exercise #14 Perform the indicated operations. a. 32 + 98 = 16 • 2 + 49 • 2 = 4 2 +7 2 = 11 2 Perform the indicated operations. b. 112 – = 28 16 • 7 – = 4 7 – 2 7 = 2 7 4 •7 Chapter 7 Section 7.3B Practice Test Exercise #16b Perform the indicated operations. b. 3 = = 3x 3 9x – 3 3x • 3 3 – 3 8 • 6x 27x = 3x – = 3x – 2 3 2 3 3 9x 2 3 6x 16 x – 48x 2 2 3 2 3x • 3 16x Chapter 7 Section 7.3B Practice Test Exercise #18b Find the product. b. = 6 + 3 6 – 3 6 + 3 6 – 3 (a + b)(a – b) = a 2 – b 2 = 6 – 3 2 = 6 – 3 = 3 2 Chapter 7 Section 7.3C Practice Test Exercise #20 Rationalize the denominator. 2 x –5 = 2 x –5 • 2 ( x + 5) = x – 25 2 x + 10 = x – 25 x +5 x +5 Section 7.4 Solving Equations Containing Radicals OBJECTIVES A Solve equations involving radicals. OBJECTIVES B Solve applications requiring the solution of radical equations. DEFINITION POWER RULE OF EQUATIONS All solutions of the equation P = Q are solutions of the n n equation P = Q , where n is a natural number. PROCEDURE TO SOLVE EQUATIONS CONTAINING RADICALS • Isolate • Repeat • Raise • Solve • Simplify • Check Chapter 7 Section 7.4A Practice Test Exercise #21 Solve. a. x +2 = –2 There is no real number solution because x + 2 ° negative number. When solving by squaring both sides: x+2=4 x=2 However, the solution x = 2 does not check. Solve. b. x = 14 x =7 x + 2 = x – 10 x +2 2 = x – 10 2 2 x + 2 = x – 20x + 100 2 0 = x – 21x + 98 0 = x – 14 = 0 or x – 14 x – 7 x –7 =0 Solve. b. Check: x = 7, x = 14 x + 2 = x – 10 ? 7 + 2 = 7 – 10 9 ° –3 x = 14, ? 14 + 2 = 14 – 10 16 = 4 The only solution is x = 14 . x =7 Chapter 7 Section 7.4A Practice Test Exercise #22 Solve. x –3 – x = –3 x –3 = x –3 x –3 2 = x – 3 2 x – 3 = x 2 – 6x + 9 2 –3 = x – 7x + 9 Solve. x =4 x –3 – x = –3 2 –3 = x – 7x + 9 2 0 = x – 7x + 12 0 = x – 4 x – 3 x – 4 = 0 or x – 3 = 0 x =3 Solve. x =4 x =3 x –3 – x = –3 Check: x = 4, ? 4–3 –4 = –3 –3 = – 3 x = 3, ? 3 – 3 –3 = – 3 –3 = – 3 The solutions are x = 4 or x = 3 . Section 7.5 Complex Numbers OBJECTIVES A Write the square root of a negative integer in terms of i. OBJECTIVES B Add and subtract complex numbers. OBJECTIVES C Multiply and divide complex numbers. OBJECTIVES D Find powers of i. DEFINITION COMPLEX NUMBER If a and b are real numbers, the following is a complex number: a + bi Real part Imaginary part RULES ADDING AND SUBTRACTING COMPLEX NUMBERS (a + bi) + (c + di) = (a + c) + (b +d)i (a + bi) – (c + di) = (a – c) + (b – d)i PROCEDURE DIVIDING ONE COMPLEX NUMBER BY ANOTHER Multiply the numerator and the denominator by the conjugate of the denominator. Chapter 7 Section 7.5A Practice Test Exercise #26b Write in terms of i. b. –98 = –1 • 49 • 2 =i •7 2 = 7i 2 Chapter 7 Section 7.5B Practice Test Exercise #27b Find. b. 5 + 2i – 7 – 8i = 5 + 2i – 7 + 8i = 5 – 7 + 2i + 8i = – 2 + 10i Chapter 7 Section 7.5C Practice Test Exercise #29b Find. b. = = 4 – 3i 3 – 5i 4 3 – 3i 3 + 5i – 5i 3 + 5i 4 3 + 4 5i + –3i 3 + –3i 5i 3 2 – 5i 2 Find. 4 – 3i 3 – 5i b. 4 3 + 4 5i + –3i 3 + –3i 5i = 3 = 2 – 5i 12 + 20i – 9i – 15i 9 – 25i 2 2 2 Find. 4 – 3i 3 – 5i b. = 12 + 20i – 9i – 15i 9 – 25i 2 2 12 + 11i + 15 = 9 + 25 27 + 11i = 34 27 11 = + i 34 34 Chapter 7 Section 7.5D Practice Test Exercise #30b Write the answer as 1, -1, i or -i. b. i – 9 = 1 9 i = 1 8 i • i = 1 1•i 1 = = = i i •i 1•i = i i 2 = –i i = –1 1 2 i • i 4