Powers, Roots and Radicals. (11) Page #23 47 Column, #51, 54, #57 73 Column, #77, 80
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1 Algebra 2/Trig Unit Notes Packet Name: Period: # Powers, Roots and Radicals () Homework Packet (2) Homework Packet () Homework Packet () Page 277 # 0 () Page #7 6 Odd (6) Page #8 60 Even (7) Worksheet evens (8) Worksheet odds (9) Page # 20 (0) Page 2 #22 79 Column () Page 2 #2 7 Column, #,, #7 7 Column, #77, 80 (2) Page 2 #2 9 Column, #2,, #9 8 Column () Chapter Review ***TEST TOMORROW***
2 . and Supplement Simplifying Radicals (Add, Sub, Conjugates no variables) (R,I,E/) Perfect Squares: A number whose square roots are integers or quotients of integers,, 9, 6, 2, 6, 9, 6, 8, 00, 2,, 69, 96, 22, 26, 289, 2, 6, (Integers) x 2, x, x 6, x 8, x 0 (Variables) /, 9/2, /9, 8/00 (Quotients of Integers) An expression with radicals is in if the following are true: ) All radicals are broken down 2) All coefficients are reduced ) No radicals are in the denominator Helpful Hint: It is often easier to break down radicals first in an attempt to make the numbers more manageable. E) Simplify the expression 0 P) Simplify the expression 8 E2) Simplify the expression a. b. 20 c. 2 0 P2) Simplify the expression a. 7 6 b. 8 c. 80 E) Simplify the expression 8wx y z 0 P) Simplify the expression 0x 8 y z 2
3 E) Simplify a b. 27 P) Simplify a b E) Simplify a. 2 8 b. 2( ) c. ( + ) 2 d. (a b)(a + b) P) Simplify a. 2 b. (7 + ) c. ( ) 2 d. (6 2)(6 + 2) E6) Simplify a. b. c d P6) Simplify a. b. + 2
4 . Complex Numbers (powers of i) (I/) Imaginary Numbers The imaginary unit i, is defined as i =. The imaginary unit i can be used to write the square root of any negative number. * Never leave an exponent other than on the imaginary unit in simplest form* i chart (cycle) i = i i 2 = i = i i = Simplify: E. 2 E2. 27 E. 8 E. 8 E. 8 8 E6. ( 6)( 0) P. 6 P2. 2 P P. 2 P. 27 P6.( 8)( 6) Complex Numbers A complex number in standard form, where a is the real part and bi is the imaginary part: a + bi E7. Write the expression as a complex number in standard form a. ( i) + ( + 2i) b. (7 i) ( i) c. 6 ( 2 + 9i) + ( 8 + i)
5 P7. Write the expression as a complex number in standard form a. ( + 2i) + ( + i) b. (2 i) ( 7i) c. 2i ( + i) + (2 i) E8. Write the expression as a complex number in standard form. a. i( 2 + i) b. (7 i)( + 2i) c. (6 + i)(6 i) P8. Write the expression as a complex number in standard form. a. i( + i) b. (2 + i)( 6 2i) c. ( + 2i)( 2i) E9. Write the quotient +i in standard form 2i 2 7i P9. Write the quotient in standard form +i E0. Solve x = -26 P0. Solve 2x = -0
6 7. nth Roots and Rational Exponents (I/2) Vocabulary: 8 8 Examples: Radical Form (Rads) Rational Exponent Form (No Rads) 2 x 2 8x 2 yz 6 7x 2 y z x 2 8 x 2 y z or 2 x 2 y z 7 6x y 2 z 6 Practice: Express using rational exponents (no rads) E. 26 E2. 6x y 6 0 E. x 6 E. 2a 0 b Express using radical notation (rads) E. x E6. 2 7a 7y 9 7 E7. a 2 g e 2 E8. 2 w 2 m 7 Simplify E9. 2 E0. 6r 6 w E. 22x y 8 E2. (2x + ) 6
7 Express in simplest radical notation (rads + simplify) E. n 7 E. x 6y 2z 7 0 E. 8 E6. (2x) 2 y 7 Evaluate (with calculator). Round answers to the nearest hundredth (2 decimal places) 8 E E8. 9 Evaluate (without calculator). The following order might help without a calculator: ) eliminate negative exp 2) change to radical form ) simplify rad ) simplify exp E9. 6 E E2. ( 6) E22. (2 2) Solve the equation (use a calculator to approximate answers). Round answers to the nearest hundredth. The following order might help ) isolate exponent 2) destroy exponent ) isolate variable (solve for x) E2. x = 2 E2. 6x = 296 E2. x = 0 E26. (x ) = 8 7
8 7.2 Properties of Rational Exponents (include examples with variables) (I/) The properties of exponents can be applied to rational exponents to simplify the expression E. Use the properties of rational exponents to simplify the expression a. 2 b. (8 2 ) 2 c. (2 ) d. 7 7 e. ( 2 ) 2 P. Use the properties of rational exponents to simplify the expression a b. (27 6 ) 2 c. ( 2 ) d. 6 6 e. ( 8 ) 9 E2. Use the properties of radicals to simplify the expression. a. 6 b P2. Use the properties of radicals to simplify the expression. a. 2 b. 2 8
9 E. Write the expression in simplest form. a. b. P. Write the expression in simplest form. a. 6 b. 7 8 E. Perform the indicated operation a. 7 (6 ) + 2 (6 ) b. 6 2 P. Perform the indicated operation a. ( ) ( ) b. 8 E. Simplify the expression. Assume all variables are positive. a. 2y 6 b. (9u 2 v 0 ) 2 c. x y 8 d. 6xy 2 2xz P. Simplify the expression. Assume all variables are positive. a. 27y 9 b. (6g h 2 ) 2 c. x y 0 d. 8rs 2 6r t 9
10 E6. Write the expression in simplest form. Assume all variables are positive. a. a b 9 c b. x y 7 P6. Write the expression in simplest form. Assume all variables are positive. a. 2d e 9 f b. g2 h 7 E7. Perform the indicated operation. Assume all variables are positive. a. y + 6 y b. 2xy 7xy c. x x 0x 2 P7. Perform the indicated operation. Assume all variables are positive. a. 8 x x b. gh 6gh c. 2 6x + x 6x 0
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