§7.1 / 7.2 Assignment Notes 1. Exercises from 7.1, 7.2 2. Nov 13 - Quiz #7: 6.6, 7.1, 7.2, Cumulative section: 6.1 - 6.3 3. Nov 20 - Exam #3: 6.1 - 6.3, 6.6, 7.1 - 7.4, 7.6; and Cumulative portion Study guide: Chapter 6 test # 1 - 12, 16 - 21; Chapter 7 test 1 - 13, 17 - 21; Cumulative Test: Chapters 5-7 #1 - 18, 23 - 31, 36 - 39 (Math 1010) M 1010 §7.1/7.2 1 / 10 §7.1 - Radicals and Rational Exponents Notes √ n-th roots of numbers ( n ) are equal value factors of a number, n-many times. Example Because 4 · 4 · 4 = 64, the number 4 is the 3rd root, or cube root, of 64. The principal root of a number has the same sign as that number. √ Example The principal square root of 49 is 7. We say 49 = 7. This avoids ambiguity because (−7) is also a square root of 49, but it is not the principal square root because it does not have the same sign as 49. (Math 1010) M 1010 §7.1/7.2 2 / 10 §7.1 - Radicals and Rational Exponents Notes When does a real number have an nth root, and how many can it have? I If real number a is positive and n is even, then a has two real nth √ √ roots, n a and − n a. I If a is any real number and n is odd, then a has only one real root, √ n a. I If real number a is negative and n is even, then a has no real nth root. Perfect squares and cubes have rational square or cube roots, respectively. This pertains also to perfect nth roots. (Math 1010) M 1010 §7.1/7.2 3 / 10 §7.1 - Radicals and Rational Exponents Notes It is useful to review page 300. Inverse properties of nth powers and nth roots: √ 1. If a has a principal root, then ( n a)n = a. √ 2. If n is odd, then n an = a. √ 3. If n is even, then n an = |a|. (Math 1010) M 1010 §7.1/7.2 4 / 10 §7.1 - Radicals and Rational Exponents a1/n = √ n Notes a Again, study page 300 and compare to page 453: 1. ar · as = ar +s ar 2. s = ar −s a 3. (ab)r = ar · b r 4. (ar )s = ars a r ar 5. = r b b 6. a0 = 1 7. a−1 = a1r a −1 b r 8. = b a (Math 1010) M 1010 §7.1/7.2 5 / 10 §7.1 - Radicals and Rational Exponents Notes √ 3 # 1: 91 Rewrite using rational exponents: x x 6 . √ 4 t # 2: 97 Rewrite using rational exponents: √ t5 # 3: 113 Simplify (c 3/2 )1/3 !3 x 1/4 # 4: 119 Simplify x 1/6 q 4 √ # 5: 123 Simplify x3 (Math 1010) M 1010 §7.1/7.2 6 / 10 §7.2 - Simplifying Radical Expressions Notes Simplyfing a radical. √ √ √ √ 12 = 4 · 3 = 4 · 3. Why do these last equalities work? There are product and quotient rules for radicals that mimic the product and quotient rules for exponents: √ √ √ 1. n u · v = n u · n v r √ n u u 2. n = √ n v v We will use this rule to remove perfect squares from radicals. First we remove constants, if able. Then we remove variable factors, if able. M 1010 §7.1/7.2 (Math 1010) 7 / 10 §7.2 - Simplifying Radical Expressions Notes Removing constants. # 1: 1 Simplify the radical. # 2: 5 Simplify the radical. √ 18. √ 96. √ # 3: 10 Simplify the radical. 1176. √ # 4: 31 Simplify the radical. 3 112. (Math 1010) M 1010 §7.1/7.2 8 / 10 §7.2 - Simplifying Radical Expressions Notes Removing variable factors. √ # 1: 19 Simplify 9x 5 . √ 3 # 2: 33 Simplify 40x 5 r 2 4 3u # 3: 52 Simplify 16v 8 (Math 1010) M 1010 §7.1/7.2 9 / 10 §7.2 - Simplifying Radical Expressions Notes Rationalizing Denominators When simplifying, we want radicals to contain no fractions, and denominators to contain no radicals. r √ √ √ √ 3 3 3 5 15 Example: =√ =√ ·√ = 5 5 5 5 5 We multiply the fraction with 1/1, written as equal factors to the denominator over equal factors, and enough to cancel out the radical. (Math 1010) M 1010 §7.1/7.2 10 / 10 §7.2 - Simplifying Radical Expressions Notes Rationalize the denominator: r 1 #1 : 55 3 s 1 #2 : 63 y #3 : 69 √ 6 3b 3 s 2x #4 : 71 3 3y (Math 1010) M 1010 §7.1/7.2 11 / 10 Notes