B. Powers of Roots

1. B. Powers of Roots Math 10: Foundations and Pre-Calculus

2. Key Terms: • Find the definition of each of the following terms: • Irrational Number • Real Number • Entire Radical • Mixed radical

3. 1. Remembering how to Estimate Roots

5. Index tells you what root to take • Radicand is what you are taking the root of

6. Construct Understanding p. 205

7. Practice • Ex. 4.1 (p. 206) #1-6

8. 2. What are Irrational Numbers? • The formulas for the are of a circle and circumference of a circle involve π, which is not a rational number • It is not rational because it cannot be expressed as a quotient of integers

9. π=3.14…..do you know what comes next? • π=3.14159265358979323846264383279….

10. What are other non-rational numbers?

11. Construct Understanding p. 207

12. Radicals that are square roots of perfect squares, cube roots of perfect squares, and so on are rational numbers • Rational numbers have decimal representations that either terminate or repeat

13. Irrational Numbers, cannot be written in the form m/n, where m and n are integers and n≠0 • The decimal representation of an irrational numbers neither terminates nor repeats

14. When an irrational number is written as a radical, the radical is the exact value of the irrational number • Examples

15. We can use the square root and cube root buttons on our calculator to determine the approximate values of these irrational numbers • Examples

16. We can approximate the location of an irrational number on a number line • If we don’t have a calculator, we use perfect powers to estimate the value

17. Example

18. Example

19. Together, the rational numbers and irrational numbers form a set of real numbers

20. Example

21. Practice • Ex. 3.2 (p. 211) #1-20 #1-2, 5-24

22. 3. What is the Difference between Mixed and Entire Radicals

26. We can use this property to simplify roots that are not perfect squares, cubes, etc, but have factors that are perfect squares, cubes, etc.

30. Example

31. Some numbers, such as 200, have more than one perfect square factor • The factors of 200 are: • 1,2,3,4,5,8,19,20,25,40,50,100,200

34. To write a radical of index n in simplest form, we write the radicand as a product of 2 factors, one of which is the greatest perfect nth power.

35. Example

39. Practice • Ex. 4.3 (p. 217) #1-22

40. 4. Fractional Exponents and Radical Relationships • Construct Understanding p. 222

41. In grade 9, you learned that for powers with variable bases and whole number exponents

42. We can extend this law to powers with fractional exponents • Example

43. So

44. Raising a number to the exponent ½ is equivalent to taking the square root of the number • Raising to the exponent 1/3 is equivalent to taking the cube root

46. Example

47. A fraction can be written as a terminating or repeating decimal, so we can interpret powers with decimal exponents. • For example, 0.2=1/5

48. So 320.2 = 321/5 • Prove it on you calculator .

49. To give meaning to a power such as 82/3, we extend the exponent law. • m and n are rational numbers