Evaluate nth Roots and Use Rational Exponents. p Evaluate nth roots and study rational exponents. VOCABULARY. Index of a radical

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1 . Georgia Performance Standard(s) MMA2a, MMA2b, MMAd Your Notes Evaluate nth Roots and Use Rational Eponents Goal VOCABULARY nth root of a p Evaluate nth roots and stud rational eponents. Inde of a radical REAL nth ROOTS OF a If n is odd, then a has real nth root. If n is even, then a has real nth roots if a > 0, real nth root if a 0, and real nth roots if a < 0. Eample Find nth roots Find the indicated real nth root(s) of a. a. n, a 264 b. n 6, a 729 Solution a. Because n is odd, 264 has. Because ( ) 264, ou can write Ï 264 or (264) /. b. Because n 6 is even and a 729 0, 729 has. Because and ( ) 6 729, ou can write 6Ï or 6729 /6. Checkpoint Find the indicated real nth root(s) of a.. n 4, a n, a 2 Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 2

2 Your Notes Eample 2 Evaluate 8 24/. Evaluate an epression with rational eponents Solution Rational Eponent Form Radical Form 8 24/ 8 24/ Checkpoint Complete the following eercise.. Evaluate (22) 22/. Eample Approimate roots with a calculator Epression Kestrokes Displa a. 4/ ^ ( ) ENTER b. 8 27/4 ^ ( ( ) ) ENTER Eample 4 Solve equations using nth roots a b. ( 4) ø 22 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

3 Your Notes Eample Use nth roots in problem solving Animal Population The population P of an animal species after t months can be modeled b P C(.2) t/ where C is the initial population. Estimate the population after 9 months if the initial population was 7. Solution P C(.2) t/ Write model for population. Substitute for C and t. ø Use a calculator. The population of the species is about after 9 months. Checkpoint Complete the following eercises. 4. Evaluate 00 /8 using a calculator. Round the result to two decimal places.. Solve ( 2 ) Homework 6. The volume of a cone is given b V πr2 h, where h is the height of the cone and r is the radius. Estimate the radius of a cone whose volume is 2 cubic inches and whose height is 6 inches. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 2

4 Name Date LESSON. Practice Rewrite the epression using rational eponent notation.. Ï 7 2. Ï 2. 4 Ï 2 4. Ï. 6 Ï 6. 8 Ï 0 7. Ï 8. 9 Ï 9. 7 Ï 9 Rewrite the epression using radical notation. 0. /2. / 2. 6 /4. 4 / /4. 7 / / 7. 6 / /2 Evaluate the epression without using a calculator. 9. Ï 20. Ï Ï Ï Ï Ï 2 2. () / 26. (227) / 27. (8) /4 24 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

5 Name Date LESSON. Practice continued Evaluate the epression using a calculator. Round the result to two decimal places when appropriate. 28. Ï Ï Ï 8. 4 Ï Ï 6. Ï 4. (48) /. (27) / 6. (8) /4 7. (7) / 8. (27) / 9. (26) /4 40. Geometr The cube below has a volume of 4 cubic inches. Find the length of an edge of the cube. 4. Basketball The volume of a basketball is approimatel cubic inches. Use the formula V 4 πr for the volume V of a sphere to estimate the radius r of the basketball. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 2

6 .2 Georgia Performance Standard(s) MMA2b Your Notes Appl Properties of Rational Eponents Goal p Simplif epressions involving rational eponents. VOCABULARY Simplest form of a radical Like radicals Radical conjugates PROPERTIES OF RATIONAL EXPONENTS Let a and b be real numbers and let m and n be rational numbers. Product of Powers Propert a m p a n a Power of a Power Propert (a m ) n a Power of a Product Propert (ab) m a b Negative Eponent Propert a 2m, a Þ 0 Zero Eponent Propert a 0, a Þ 0 Quotient of Powers Propert a m a n a, a Þ 0 Power of a Quotient Propert a b 2 m, b Þ 0 26 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

7 Your Notes Eample Use properties of eponents Use the properties of rational eponents to simplif the epression. a. 9 /2 p 9 /4 b. (7 2/ p /6 ) c. /6 / d. 6 2/ 4 2/ 2 4 PROPERTIES OF RADICALS Product Propert of Radicals Quotient Propert of Radicals n Ï a p b nî a b, b Þ 0 Eample 2 Use properties of radicals Use the properties of radicals to simplif the epression. a. Ï 27 p Ï 9 b. Product propert Ï 92 Ï Quotient propert Checkpoint Simplif the epression.. (6 6 p 6 ) 2/6 2. Ï 24 Ï Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 27

8 Your Notes Eample Write a radical in simplest form Write Ï 28 in simplest form. Solution Ï 28 Factor out perfect fifth power. p Product propert Simplif.. Checkpoint Write the epression in simplest form. 4 Ï Î 9 Eample 4 Simplif the epression. a. 2(2 2/ ) 7(2 2/ ) b. Add and subtract like radicals and roots 4 Ï 48 2 Ï 4 p 2. Checkpoint Simplif the epression. Ï 20 2 Ï Ï Ï 6 28 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

9 Your Notes Eample Simplif epressions involving variables Simplif the epression. Assume all variables are positive. a. Ï 2 b. (6m 4 n 0 ) /2 c. Î a 9 b 6 d. h 2 Ï t Eample 6 Add and subtract epressions involving variables Perform the indicated operation. Assume all variables are positive. a. 0Ï 2 6 Ï b. a 2 b /4 4a 2 b /4 Checkpoint Simplif the epression. Assume all variables are positive. 7. Ï 8 7 z 8. 7 Ï 2a 2 aï 28a 2 Homework Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 29

10 Name Date LESSON.2 Practice Simplif the epression using the properties of rational eponents.. /2 p /2 2. ( 2/ ) /2. (4 p ) 2/ /2. 9 / 9 2/ /6 Simplif the epression using the properties of radicals. 7. Ï p Ï 2 8. Ï 4 p Ï Ï 27 Ï 0. Î Ï p 4 Ï Ï 0 p Ï 2 Ï Simplif the epression. Assume all variables are positive.. / p 4/ 4. ( 2/ ) 2. ( /2 ) /2 6. (8) / 7. 24/ 8. ( /6 ) 2 9. /6 / / / / 0 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

11 Name Date LESSON.2 Practice continued Perform the indicated operation. Assume all variables are positive. 22. Ï Ï 2. 6 Ï 7 2 Ï Ï Ï 2. 7 Ï 2 2 Ï Ï 27 4 Ï Ï 6 Ï Ï 2 8 Ï 29. Ï 2 Ï Ï 2 6 Ï 4 4 Write the epression in simplest form. Assume all variables are positive.. Ï Î 4. Ï 4 z 4. Ï 8 6 z 4. Ï 2 z Ï 9 2 z 6. Ï 2 z 7 Ï Ï 8. 4 g 2 Ï f 9. Geometr The area A of an equilateral triangle is given b A Ï 4 s2 where s is the length of each side. Find the area of an equilateral triangle with a side length of 64 /4 units. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics

12 . Graph Square Root and Cube Root Functions Georgia Performance Standard(s) MMAd Your Notes Goal p Graph square root and cube root functions. VOCABULARY Radical functions Eample Graph a square root function Graph Ï, and state the domain and range. Compare the graph with the graph of Ï. Solution Make a table of values and sketch the graph The domain is 0. The range is 0. The graph of Ï is a vertical of the parent graph of Ï. Checkpoint Complete the following eercise.. Graph 2Ï. Then state the domain and range. 2 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

13 Your Notes Eample 2 Graph a cube root function Graph 2 Ï, and state the domain and range. Compare the graph with the graph of Ï. Solution Make a table of values and sketch the graph The domain and range are. The graph of 2 Ï is a vertical of the parent graph of Ï b a factor of. Checkpoint Complete the following eercise. 2. Graph 2 Ï. Then state the domain and range. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics

14 Your Notes Eample Football A football falls out of a tree from a height of 40 feet. The time t (in seconds) it takes for the football to fall to a height of h feet is given b the function t 4 Ï 40 2 h. Solve a multi-step problem a. Use a graphing calculator to graph the model. b. What is the height of the football after 4 seconds? Round our answer the nearest foot. Solution a. Graph the model Enter. Use a viewing window of and. b. Trace to find the -coordinate when. The graph shows ø. The height of the football after 4 seconds is about. X= Y= Checkpoint Complete the following eercise.. Rework Eample to find the height of the football after 2 second. Round our answer to the nearest foot. 4 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

15 Your Notes Eample 4 Graph a translated radical function Graph 22Ï 2 2. Then state the domain and range. Solution. Sketch the graph of 22Ï. Notice that it passes through (, ), (, ), and (, ). 2. Note that for 22Ï 2 2, h and k. So, shift the graph and. The resulting graph passes through the points (, ), (, ), and (, ). From the graph, ou can see that the domain and range of the function are both. 2 2 Checkpoint Graph the function. Then state the domain and range Ï 2. Ï Homework Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics

16 Name Date LESSON. Practice Match the function with its graph.. f() Ï 2. f() Ï. f() 2 Ï 4. f() Ï. f() Ï 2 6. f() 2 Ï A. B. C. 2 2 D. E. F. 2 6 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

17 Name Date LESSON. Practice continued Graph the function. Then state the domain and range. 7. f() 2 Ï 8. f() Ï 9. f() Ï 2 0. f() 2 Ï 2. f() Ï 2 2. f() Ï 2 In Eercises and 4, use the following information. Falling Object A stone is dropped from a height of 0 feet. The time it takes for the stone to fall to a height of h feet is given b the function t 4 Ï 0 2 h where t is the time in seconds.. Identif the domain and range of the function. 4. What is the height of the stone after 2 seconds? Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 7

18 .4 Solve Radical Equations Georgia Performance Standard(s) MMAd Your Notes Goal p Solve radical equations. VOCABULARY Radical equation Eample Solve a radical equation Solve Ï 6. Solution Ï 6 Write original equation. Square each side to eliminate the radical. The solution is Simplif. Subtract from each side.. Check this in the original equation. Checkpoint Solve the equation. Check our solution.. Ï Ï Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

19 Your Notes Eample 2 Solve a radical equation given a function Speed of Sound in Air The speed S of sound (in meters per second) is given b S Ï 286HT where H is the ratio of specific heats and T is the temperature (in Kelvin). Find the temperature when S 40 meters per second and H.4. Solution S Ï 286HT Write given function. Substitute for S and for H. Simplif. Square each side. Simplif. ø Divide each side b. The temperature is about Kelvin. Eample Solve an equation with a rational eponent Solve ( 4) /2 64. Solution ( 4) /2 64 Write original equation. Raise each side to the power 2. Appl properties of eponents. Simplif. Subtract from each side. Divide each side b. The solution is. Check this in the original equation. Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 9

20 Your Notes Eample 4 Solve an equation with two radicals Solve Ï 6 2 Ï 0 2. Ï 6 2 Ï 0 2 Write original equation. Square each side. Epand left side and simplif right side. Isolate radical epression. Divide each side b 4. Square each side again. 0 or 0 Simplif. 0 Write in standard form. 0 Factor. Zero product propert or Solve for. CHECK Check in the original equation. 0 0 Þ Check in the original equation. 0 0 The onl solution is is etraneous.). (The apparent solution 40 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

21 Your Notes Checkpoint Complete the following eercises.. Rework Eample 2 when S 9 meters per second. 4. Solve 22 / Check for etraneous solutions.. Solve Ï 4 2 Ï. Check for etraneous solutions. Homework Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 4

22 Name Date LESSON.4 Practice Check whether the given -value is a solution of the equation.. Ï 7; 6 2. Ï 6; 8. Ï 2 ; 8 4. Ï ; 222. Ï 2 0; 7 6. Ï ; 4 Solve the equation. Check our solution. 7. Ï 6 8. Ï 7 9. Ï 2 0. /2 2. Ï Ï Ï Ï ( 2 ) /2 Solve the equation. Check our solution. 6. Ï Ï Ï ( 4) / Ï / Ï ( ) / Ï Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.

23 Name Date LESSON.4 Practice continued Solve the equation. Check for etraneous solutions. 2. 2/ Ï Ï Ï 2 Ï 28. Ï Ï / Ï 2 Ï. Ï Ï 2 4 Ï 2 8. Ï 2 In Eercises 4 6, use the following information. Pendulum The period of a pendulum is the time T (in seconds) it takes for a pendulum of length L (in feet) to go through one ccle. The period is given b T 2π Î L. Given the 2 period of a pendulum, find its length. Round our answers to two decimal places. 4. T. seconds. T 2. seconds 6. T seconds In Eercises 7 9, use the following information. Velocit The velocit of a free falling object is given b V Ï 2gh where V is velocit (in feet per second), g is acceleration due to gravit (in feet per second squared), and h is the distance (in feet) the object has fallen. On Earth g 2 ft/s 2. Given the velocit of the object when it hits the ground, find the distance from which it was dropped. 7. V 72 ft/s 8. V 04 ft/s 9. V 60 ft/s Copright McDougal Littell/Houghton Mifflin Compan. Georgia Notetaking Guide, Mathematics 4

24 Words to Review Give an eample of the vocabular word. nth root of a Inde of a radical Simplest form of a radical Like radicals Radical conjugates Radical function Radical equation 44 Georgia Notetaking Guide, Mathematics Copright McDougal Littell/Houghton Mifflin Compan.