Module 6 Exponents and Radicals In this module, rules are extended to include all integral as well as rational exponents. Zero and Negative Exponents If a is a real non- zero number, a0 =1an= 1 n a Examples: 1. 2000 = 1 2. ( x4y2 )0 = 1 3. 2 ( a3b4) 0 = 2 1 3 x = 3 4. x 5. 2 ab c = ab c2 Laws of Exponents If a, b are real numbers and m, n are any integers, then m n m +n I. a a =a II. ( a m ) =a mn III. ( ab )m =am am n a m am = m b b IV. () V. am =a mn n a Examples: 52 53=5 5 1. 2. 5 5 (x ) x5 x = = 5 5 y y ( y) ( ) 2 3. 2x x + y1 1 1 ( x ) = = 2 2 1 1 + x y 2 x2 2 xy 2y = 2 = y + x x y + x x ( y+ x ) xy The Radical Notation If a and b are real numbers such that an = b then a is called an nth root of b. Any number has exactly n nth roots. One of the nth roots of a number is its principal roots. n b stands for the principal nth root of b. The principal nth root of a positive number is its positive root. If b is negative and n is odd, the principal nth root of b is the negative root. If b is negative and n is even, then b has no real roots. In the symbol n b , b is called the radicand, n is the index and \" \" is the radical sign. The expression Examples: 1. 64=8 3 2. 125=5 n b is a radical. 5 3. 32=2 4 4. 16 is not real Rational and Fractional Exponents If a is any real number and m, n are positive integers, then n n m n a1 /n = aa m /n= a m=( a ) Examples: 3 1. 4 2 = 4 = 64=8 2. 1252/ 3= ( 125 ) =( 125 ) =52=25 3. 3 3 163 /4 = 2 3 2 1 1 1 1 = 4 = 3= 3/ 4 3 16 ( 16 ) 2 8 Laws of Radicals Simplifying expressions containing radicals uses the following laws of radicals. For any real numbers a, b and positive integers m, n. I. n an II. n ab= n a n b III. IV. Examples: n = n a a = b n b n m a= a mn n ( n a ) =a 3 3 3 3 1. 64 x = ( 4 x ) =4 x 2. 2 10= 20= ( 4 ) ( 5 )= 4 5=2 5 3. 4 4 a8 a 8 a2 = = x 4 4 x 4 x 6 3 3 4. 25= 25= 5 Simplification of Radicals A radical is in its simplest form if: 1. The radicand contains no factor raised to a power equal to or greater than the index of the radical. 2. No radical appears in the denominator of a fraction or no fraction is within a radical. 3. The exponent of the radicand and the index of the radical have no common factor except 1. Examples: 4 3 4 2 1. 60 x y = ( 4 x y ) (15 y ) ( 4 x y ) 15 y 4 2 2 x 2 y 15 y 2. 3 2 2 5 x2 =3 25 x 25 x 5 x 2 3 2 10 x 3 125 x 3 10 x 2 5x 3 9 3 9 3. 27 x y = 27 x y 6 3 3 xy 3 y 3 xy Addition and Subtraction of Radical Expressions An algebraic expression containing radicals is called a radical expression. Similar radical expressions have the same index and the same radicand. Similar radical expressions can be combined. To add radical expressions, simplify each expression and then combine similar radicals. Example: 15 2010 12+2 75 15 ( 4 ) ( 5 )10 ( 4 ) (3 )+ 2 ( 25 ) ( 3 ) ( 15 ) (2 ) 5( 10 )( 2 ) 3+ ( 2 )( 5 ) 3 30 520 3+ 10 3 30 510 3 Multiplication of Radical Expressions Radical expressions having the same index are multiplied by multiplying the radicands. If radicals are of different orders, change first to radicals of the same order. Examples: 4 4 3 1. 2 y 16 y 4 32 y 4 = 4 ( 16 y ) ( 2 ) = 2 y 4 2 = 3 4 2. 3 2 = = = 4 3 1/3 2 3 4/12 2 3/12 12 34 23 12 81 8 = 12 648 = 3. ( 2 3 ) ( 2 23 ) = 2 42 6 6+ 9 = 2 (2 )3 6 +3 = 73 6 Division of Radical Expressions Radical expressions having the same index are divided by dividing the radicands. If radicals are of different indices, change them first to radicals with a common index. Examples: 1. 3 16 = 3 16 = 3 8 2 3 2 = 2 3 1/2 1/ 3 2. 3 2=3 2 = = = = = 3. 24 6 = 24 6 2 2 2 = 12 3 = 2 3 3 = 3 3 /6 2 3 3 3 2 2 6 6 6 2 /6 33 24 22 24 432 6 2 6 432 2 Rationalizing Denominators When the denominator of a fraction containing radicals is a binomial which contains a second radical, the denominator is rationalized by multiplying both numerator and denominator by a property chosen radical expression. Example: 2 2 x+ y = x+ y x+ y x+ y = = 2 x + y ( x + y )2 2 x+ y x+ y Assignment 6 A. Simplify each expression leaving the result without zero or negative exponents. 1. [ ( p2 q )4 ] ( pq ) 2 3 2 2. ( 3 x y ) 3 0 3. a4 b2 c 6 a2 b3 c 4 4. ( x+2 y )5 12 ( x +2 y ) 5. p2 q3 r4 p 2 q 5 r 1 ( 4 ) 1 ( 40 x4 ) 6. ( 4 x 0 y3 )2 7. x 4 a x2 a xa 8. ( 3 ab )2 ( 2 a3 b0 ) 9. x 2 x x 3 + x 1 3 ( 3 ) ( x2 y5 y 1 ) 10. 1 B. Evaluate: 4 1 1. ( 3 ) 2. ( 81 )2/ 4 3. ( 32 )3/ 5 4. 9 16 3 /2 ( ) 3/ 2 2 5. ( 64 ) C. Express the following radicals in simplest forms. 4 3 1. 25 x y 2. 25 15 12 x y 64 3 8 10 3. 54 x y 3 4 22 4. 64 x y 4 4 8 5. 128 a b 4 6. 4 7. a 4 b3 4 2 x y 3 4 2 2 8. a +2 ab+b 6 4 2 9. a 4 a +4 a 10. 4+ 4 1 + 4 2 y y D. Combine into a single radical. 1. 50+ 48+ 75 2. 4 2010 75+ 6 12 3. 4. 8 3 10824 3 32 1 3 +6 3 27 4 3 3 6 5. 3210 46 16 6. x 5 x2 y 20 x 5 y + 4 x 2 45 y 4 7. 8. a 2 b 2 + ab a 1 b ab 75 b 4b 25 c + + c 3c b 6 4 9 6 2 2 9. m n m n mn 10. 2 3 x+ y 3 x y 9x + 2 2 3 x y 3 x+ y 9x y E. Multiply or divide as indicated. 1. 30 15 3 2 3 2 2. 8 x 3 xy 3 2 4 3. a a 3 4. ( 53 7 ) 5. ( 6+3 2 )( 2 64 2 ) 2 2 6. ay a y 7. 4 100 100 z 4 2 4 200 3 8. 2 2 3 9. 9 3 10. 16 x 4 3 2 x F. Simplifying the following by rationalizing the denominator. 6 x 3 18 x 2 1. 3 x 2. 2+ 7 2 7 3. 10 2 x 10+ 2 x 4. 3 11 3 2 11+ 3 a+ b ab 5. a+b+ a - b -End