Roots and Powers

1. Roots and Powers Hughes, Cynthia, Cherie, Scarlett

2. 1 Review Time 2 Irrational Number Mixed and Entire Radicals Fractional Exponents & Radicals 3 Negative Exponents, Reciprocals, and the Exponent Laws Root Power & 4

3. Part 1 Irrational Number By hughes

4. Classifying and Ordering Numbers IRRATIONAL NUMBERS

5. Rational numbers are numbers that can be written in the form of a fraction or ratio, or more specifically as a quotient of integers • Any number that cannot be written as a quotient of integers is called an irrationalnumber • ∏ is one example of an irrationalnumber…. • √0.24, 3√9, √2, √1/3, 4√12, e • Some examples of rationalnumbers? • √100, √0.25, 3√8, 0.5, 5/6, 7, 5√-32

6. Can you think of any more? Rational numbers: Irrationalnumbers:

7. Rational Vs. Irrational Numbers • You should have noticed that the decimal representation of a rational number either terminates, or repeats • 0.5, 1.25, 3.675 • 1.3333…., 2.14141414….. • The decimal representation of an irrational number neither terminates nor repeats • 3.14159265358………..

8. So…………………………………………………. • Which of these numbers are rational numbers and which are irrational numbers? • √1.44, √64/81, 3√-27, √4/5, √5 R R R I I

9. Exact Values Vs. Approximate Values • When an irrational number is written as a radical, for example; √2 or 3√-50, we say the radical is the exact valueof the irrational number. • When we use a calculator to find the decimal value, we say this is an approximate value • We can approximate the location of an irrational number on a number line

10. Summary Of Number Sets

11. Example 1 Order these numbers on a number line from least to greatest: 3√13, √18, √9, 4√27, 3√-5 3√13 ≈ 2.3513… √18 ≈ 4.2426… √9 = 3 4√27 ≈ 2.2795… 3√-5 ≈ -1.7099… From least to greatest: 3√-5, 4√27, 3√13, √9, √18

12. Bones

13. Part 2 Mixed and Entire Radicals By Cherie

14. Example 1 Simplify the radical √80 Simplifying Radicals Using Prime Factorization • Solution: • √80 = √8*10 • = √2*2*2*5*2 • = √(2*2)*(2*2)*5 • = √4*√4*√5 • =2*2*√5 • =√5

15. Multiple Answers • Some numbers, such as 200, have more than one perfect square factor • The factors of 200 are: 1,2,4,5,8,10,20,25,40,50,100,200 • Since 4, 25, and 100are perfect squares, we can simplify √200 in three ways: • 2√50, 5√8, 10√2 • 10√2 is in simplest form because the radical contains no perfect square factors other than 1.

16. Example 2) Writing Radicals in Simplest Form • Write the radical in simplest form, if possible. • 3√40 • Solution: • Look for the perfect nth factors, where n is the index of the radical. • The factors of 40 are: 1,2,4,5,8,10,20,40 • The greatest perfect cube is 8 = 2*2*2, so write 40 as 8*5. • 3√40 = 3√8*5 = 3√8*3√5 = • 23√5 • Your turn: • Write the radical in simplest form, if possible. • √26, 4√32 • Cannot be simplified, 24√2

17. Mixed and Entire Radicals • Radicals of the form n√x such as √80, or 3√144 are entire radicals • Radicals of the form an√x such as 4√5, or 23√18 are mixed radicals • (mixed radical  entire radical)

18. Example 3) Writing Mixed Radicals as Entire Radicals • Write the mixed radical as an entire radical • 33√2 • Solution: • Write 3 as: 3√3*3*3 = 3√27 • 33√2 = 3√27 * 3√2 = 3√27*2 = • 3√54 • Your turn: • Write each mixed radical as an entire radical. • 4√3, 25√2 • √48, 5√64

19. Review • Multiplication Property of Radicals is: • n√ab = n√a * n√b, where n is a natural number, and a and b are real numbers • 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 • Radicals of the form n√x such as √80, or 3√144 are entire radicals • Radicals of the form an√x such as 4√5, or 23√18 are mixed radicals

20. Part 3 Fractional Exponents and Radicals By Cynthia

21. Powers with rational exponents with numerators 1 When n is a natural number and x is a rational number!!!!!!

22. Examples

23. Tryit Evaluate each power without using a calculator: 271/3 271/3 = 3√27 = 3 0.491/2 0.491/2 = √0.49 = 0.7

24. Powers with Rational Exponents When m and n are natural numbers, and x is a rational number: xm/n = (x1/n)m = (n√x)m And: xm/n = (xm)1/n = n√xm

25. TRY IT Write 402/3 in radical form in 2 ways ANSWER: Use am/n = (n√a)m or n√am 402/3 = (3√40)2 or 3√402

26. Review Powers with Rational Exponents with Numerator 1 When n is a natural number and x is a rational number: x1/n = n√x Powers with rational exponents When m and n are natural numbers, and x is a rational number: xm/n = (x1/n)m = (n√x)m And: xm/n = (xm)1/n = n√xm

27. Part 4 Negative Exponents, Reciprocals, and the Exponent Laws By Scarlett

28. Basic Reciprocals 4 x ¼ = 1 2/3 x 3/2 = 1 Any two numbers that have a product of 1 are calledreciprocals Also Applies to Powers am x an = am+n

29. Basic Laws of Exponents When x is any non-zero number and n is a rational number, x-n is the reciprocal of xn x-n = (1/x)n (1/x)-n = xn, x ≠ 0 Product of Powers: am x an = am+n Quotient of Powers: am/an = am-n, a ≠ 0 Power of a Power: (am)n = amn Power of a Product: (ab)m = ambm Power of a Quotient: (a/b)m = am/bm, b ≠ 0

30. Example Example 1 • Solution: • According to the Law: • x-n = (1/x)n • 3-2 = (1/3)2 • 1/9 Evaluate the power below: 3-2

31. Example Example 2 • Solution: • According to the Law: • (1/x)-n = xn, x ≠ 0 • (-3/4)-3 = (-4/3)3 • -64/27 Evaluate the power below: (-3/4) -3

32. Example • Solution: • 8-2/3 = (1/8)2/3 = 1/(3√8)2 • (1/2)2 • ¼ • x-n = (1/x)n • (1/x)-n = xn, x ≠ 0 Example 3 Evaluate the power below: 8-2/3

33. Example Example 4 • First use the quotient of powers law • 4/2 x a-2/a2 x b2/3/b1/3 = 2 x a(-2)-2 x b2/3-1/3 • 2a-4b1/3 • Then write with a positive exponent • 2b1/3/a4 Simplify the expression 4a-2b2/3/2a2b1/3

34. Your Turn Evaluate the power below: (Choose only 2 of them) 1) (9/16)-3/2 2) (7/24) -1/9 3) (9/20) 7/4 4) (25/10) -1/3

35. Thank Made by Math Group#6 Scarlett, Cherie, Cynthia and Hughes You