4.3 Mixed and Entire Radicals

Transcription

1 4.3 Mixed and Entire Radicals

2 Index Review of Radicals Radical 3 64 Radicand When the index of the radical is not shown then it is understood to be an index of = 2 64

3 MULTIPLICATION PROPERTY of RADICALS What do you notice?

4 MULTIPLICATION PROPERTY of RADICALS n ab = n a n b where n is a natural number, and a and b are real numbers

5 Example 1: 24 = 4 6 = 2 6 = 2 6

6 Example 2: 3 24 = = = 2 3 3

7 Simplify each radical. Write each radical as a product of prime factors, then simplify. Since 80 is a square root. Look for factors that appear twice

8 Simplify each radical. Write each radical as a product of prime factors, then simplify. Since 144 is a cube root. Look for factors that appear three times

9 Simplify each radical. Write each radical as a product of prime factors, then simplify. Since 162 is a fourth root. Look for factors that appear four times

10 PRACTICE PROBLEM Simplify each radical.

11 Mixed Radical: the product of a number and a radical 4 6 Entire Radical: the product of one and a radical 72

12 Writing Mixed Radicals as Entire Radicals n ab = n a n b n a n b = n ab

13 Write each mixed radical as an entire radical n a n b = n ab

14 PRACTICE PROBLEMS Write each mixed radical as an entire radical

15 Work in groups of two The first group to finish wins a prize!

16 Simplify the radicals around the edges of the individuals squares Then cut out their pieces and rearrange them so that their edges match Paste them onto a blank template given

17 Simplify the radicals around the edges of the individuals squares Then cut out their pieces and rearrange them so that their edges match Paste them onto a blank template given

18 Simplify the radicals around the edges of the individuals squares Then cut out their pieces and rearrange them so that their edges match Paste them onto a blank template given

19

20 4.4 Fractional Exponents and Radicals

21 REMEMBER Grade 9? a m BASE

22 REMEMBER Grade 9? a m EXPONENT

23 REMEMBER Grade 9? a m a n = a m+n We can further use it to calculate fractional exponents with numerator 1

24 WHAT IS A FRACTIONAL EXPONENT? a x y

25 A FRACTIONAL EXPONENT with a numerator 1 a 1 y

26 5 1 2 = = = = = 25 = and 5 equivalent expressions

27 This suggests = = 3 5 x 1 n = n x

28 Powers with Rational Exponents = = 3 5 x 1 n = n x When n is a natural number and x is a rational number,

29 Evaluate each power without using a calculator = 3 27 = = 0.49 = 0.7 ( 4 9 )1 2 = 4 9 = 2 3

30 PRACTICE PROBLEMS Evaluate each power without using a calculator

31 What if. The numerator in the exponent IS NOT 1? x 1 n = n x??

32 RECALL.. (a m ) n = am n So, for example,

33 8 2 3 = (a m ) n = am n = (8 1 3) 2 am n = (a m ) n = ( 3 8) 2 But, this is also true = (2) 2 = 4

34 = (a m ) n = am n am n = (a m ) n =(8 2 ) 1 3 = But, this is also true = 3 64 = 4 x 1 n = n x

35 Powers with Rational Exponents When m and n are natural numbers and x is a rational number, m 1 x n =(xn) m m x = n (x m 1 ) n =( n x) m = n x m

36 a. Write in radical form in 2 ways b. Write 3 5 and ( 3 25) 2 in exponent form = ( 3 40)² and 3 40² x m n =(x 1 n) m =( n x) m 3 5 = x m n = (x m ) 1 n ( 3 25) 2 = = n x m

37 a) Write in radical form in 2 ways b) Write 6 5 and ( 4 19) 3 in exponent form.

38 Biologists use the formula b = 0.01m 2 3 to estimate the brain mass, b kilograms, of a mammal with body mass m kilograms. Estimate the brain mass each animal a. A husky with a body mass of 27 kg b. A polar bear with a body mass of 200g

39 Biologists use the formula b = 0.01m 2 3 to estimate the brain mass, b kilograms, of a mammal with body mass m kilograms. Estimate the brain mass each animal a. A husky with a body mass of 27 kg b. A polar bear with a body mass of 200g Substitute: m = 27 b = 0.01(27) 2 3 b = 0.01( 27) 2 b = 0.01(3) 2 b = 0.09 kg The brain mass of the husky is approximately 0.09 kg.

40 Biologists use the formula b = 0.01m 2 3 to estimate the brain mass, b kilograms, of a mammal with body mass m kilograms. Estimate the brain mass each animal a. A husky with a body mass of 27 kg b. A polar bear with a body mass of 200g Substitute: m = 200 b = 0.01(200) 2 3 USE a CALCULATOR! The brain mass of the polar bear is approximately 0.34 kg.

41 POWERPOINT PRACTICE PROBLEM Use the formula b = 0.01m 2 3 to estimate the brain mass of each animal. A moose with a body mass of 512 kg A cat with a body mass of 5 kg

42 4.5 Negative Exponents and Reciprocals

43 What is a RECIPROCAL? Two numbers with a product of 1 are RECIPROCALS. For example: = = 1

44 Without a calculator, Calculate = 1 Reciprocals!! 5 2 = = They are RECIPROCALS of each other!!

45 5 2 = = x n = 1 x n x n = 1 x n

46 Evaluate each power without using a calculator (a and b only) x n = 1 x n x n = 1 x n

47 PRACTICE PROBLEM Evaluate each power without using a calculator (a and b only)

48 Evaluate each power without using a calculator x m n = (x m ) 1 n x m n =(x 1 n) m = n x m =( n x) m x n = 1 x n x n = 1 x n

49 Evaluate each power without using a calculator

50 PRACTICE PROBLEM 2 Evaluate each power without using a calculator

51 Paleontologists use measurements from fossilized dinosaur tracks and the formula v = 0.155s 5 3f 7 6 to estimate the speed at which the dinosaur travelled. In the formula, v is the speed in metres per second, s is the distance between successive footprints of the same foot, and f is the foot length in metres. Use the measurements in the diagram to estimate the speed of the dinosaur.

52 Paleontologists use measurements from fossilized dinosaur tracks and the formula v = 0.155s 5 3f 7 6 to estimate the speed at which the dinosaur travelled. In the formula, v is the speed in metres per second, s is the distance between successive footprints of the same foot, and f is the foot length in metres. Use the measurements in the diagram to estimate the speed of the dinosaur. The dinosaur travelled at approximately 0.8 m/s.

53 Use the formula v = 0.155s 5 3f 7 6 to estimate the speed at which the dinosaur travelled when s = 1.5 and f = 0.3. The dinosaur travelled at approximately 1.2 m/s.

54 4.6 Applying the Exponent Laws

55 a m a n = a m+n a m a n = a m n (a m ) n = am n (ab) m = a m b m ( a b )m = am b m

56 Simplify by writing as a single power. Explain the reasoning. a) = = a m a n = a m+n b) (1.43 )(1.4 4 ) (1.4 2 ) = = = ( 2) = c) ( 72 3 a m a n = a m+n ) 6 = a m a n = a m n

57 Simplify by writing as a single power. Explain the reasoning. c) ( ) 6 = ( ) 6 = ( ) 6 = (7 4 3) 6 = = 7 8 = a m a n = a m+n a m a n = a m n (a m ) n = am n

58 PRACTICE PROBLEM Simplify by writing as a single power. Explain the reasoning.

59 Simplify. Explain the reasoning. a) (x 3 y 2 )(x 2 y 4 ) b) 10a5 b 3 2a 2 b 2

60 Simplify. Explain the reasoning. a) (x 3 y 2 )(x 2 y 4 ) = x 3 y 2 x 2 y 4 a m a n = a m+n = x 3 x 2 y 2 y 4 = x 3+2 y 2+( 4) = x 5 y 2 = x5 y 2

61 Simplify. Explain the reasoning. 10a 5 b 3 2a 2 b 2 = 10 2 a5 b3 a2 b 2 = 5 a 5 2 b3 2 = 5 a 3 b 5 = 5a 3 b 5

62 PRACTICE PROBLEM Simplify. Explain the reasoning.

63 Simplify. Explain the reasoning.

64 Simplify. Explain the reasoning.

65 Simplify. Explain the reasoning.

66 Simplify. Explain the reasoning.

67 Simplify. Explain the reasoning.

68 PRACTICE PROBLEM Simplify. Explain the reasoning.

69 A sphere has a volume 425 m³. What is the radius of the sphere to the nearest tenth of a metre? 4 3 πr³ 425 = 4 3 πr³ 1275 = 4πr³ 1275/4π = r³ ( π )1 3 = r r

70 PRACTICE PROBLEM A cone with height and radius equal has volume 18 cm 3. What are the radius and height of the cone to the nearest tenth of a centimetre?