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Please CLOSE YOUR LAPTOPS, and turn off and put away your cell phones, and get out your notetaking materials. Math 110 Final Exam: • • • • • Comprehensive – covers the whole semester Worth 200 points (20% of course grade) 45 questions Test period is one hour and 50 minutes Practice Final will be worth 10 points and also has 45 questions and unlimited tries. • Your best score on the practice final will also earn up to 20 extra credit points on the final. (more details next week) Make sure you know the day and time of the final exam for this section of Math 110: • All Math 110 finals will be given in your regular classroom. • (Next slide shows final exam schedules for all sections.) Monday 5/2 Tuesday 5/3 Scheduled Final in 214: Scheduled Final in 214: Scheduled Final in 214: 110-001 Neiderhauser 8:00 110-002 Wojciechowski 10:10 110-004 Lee 2:30 8:00 to 10:00 Wednesday 5/4 Thursday 5/5 Friday 5/6 LAB CLOSED OPEN LAB in 203: OPEN LAB in 203: OPEN LAB in 203: OPEN LAB in 203: Scheduled Final in 214: Scheduled Final in 214: OPEN LAB in 203: Scheduled Final in 214: 110-005 Corne 3:35 010-001 Schmidt MW 9:05 10:00 to 12:00 110-003 Lee 1:25 12:00 to 2:00 LAB CLOSED Scheduled Final in 214: OPEN LAB in 203: OPEN LAB in 203: 010-002 Schmidt TTh 9:05 2:00 to 4:00 LAB CLOSED OPEN LAB in 203: 4:00 to 6:00 OPEN LAB in 203: OPEN LAB in 203: Section 10.2 Radicals and Rational Exponents Definition of a rational exponent in terms of a radical: If n is a positive integer greater than 1 and a is a real number, then 1/ n a a n Why does this definition make sense? Recall that a cube root is defined so that 3 a b only if b 3 a However, if we let b = a1/3, then b3 (a1/ 3 )3 a 1/ 33 a1 a Since both values of b give us the same a, a1/ 3 3 a Example Use radical notation to write the following. Simplify if possible. 811/ 4 32x 10 1 / 5 4 81 4 34 3 5 32 x10 5 25 x10 2 x 2 We can expand our use of rational exponents to include fractions of the type m/n, where m and n are both integers, n is positive, and a is a positive number, a m/n a n m a n m Example Use radical notation to write the following. Simplify if possible. 8 4/3 8 3 4 2 3 3 4 24 16 Problem from today’s homework: 64 Now to complete our definitions, we want to include negative rational exponents. If a-m/n is a nonzero real number, a m / n 1 a m/n Example Use radical notation to write the following. Simplify if possible. 64 16 2 / 3 5 / 4 1 2/3 64 1 64 2 3 1 5/ 4 16 2 4 4 3 1 4 1 5 3 2 1 1 2 16 4 1 1 5 2 32 5 / 4 ( 16 ) What if the previous problem was ? The answer would be “N” (not a real number) because you’d be trying to take an even root of a negative number. All the properties that we have previously derived for integer exponents hold for rational number exponents, as well. We can use these properties to simplify expressions with rational exponents. Example Use properties of exponents to simplify the following. Write results with only positive exponents. 32 1/ 5 x 2/3 3 32 3/ 5 x 2 2 x 5 5 3 2 3 2 2 2 x 8x a1/ 4 a 1/ 2 1 1 / 4 1 / 2 2 / 3 3 / 12 6 / 12 8 / 12 11/ 12 a a a 11/12 2/3 a a 𝟏 What would this answer look like in radical form? 𝟏𝟐 𝒂𝟏𝟏 Problem from today’s homework: Final answer: -b Hint: The exponent will be 2/5 + 1/5 – (-2/5). Then simplify the fraction. Example Use rational exponents to write as a single radical. 3 5 2 5 2 1/ 3 1/ 2 5 2/6 2 3/ 6 5 2 2 3 1/ 6 6 200 Problem from today’s homework: Final answer: 20 y19 Hints: Start by writing each radical as a rational (fraction) exponent, then add the fractions by finding a common denominator. For your final step, convert back into radical form. The assignment on this material (HW 10.2) is due at the start of the next class session. You may now OPEN your LAPTOPS and begin working on the homework assignment until the end of the class period.