1. 1 P.2 INTEGER AND RATIONAL NUMBER EXPONENTS Objectives:  Properties of Exponents  Scientific Notation  Rational Exponents and Radicals  Simplifying Radical Expressions ( الاسس الصحيحة والنسبية )

2. 2 Def: If a is a real number and n is a positive integer, then Ex:

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4. 4 Laws of Exponents LawExample

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6. 6 Ex: Simplify the following expression

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8. 8 More Examples

9. 9 Converting a Decimal to Scientific Notation 1. Count the number N of places that the decimal point must be moved in order to arrive at a number x, where 1 < x < 10. 2. If the original number is greater than or equal to 1, the scientific notation is 3. If the original number is between 0 and 1, the scientific notation is

10. 10 Ex : Write the number 5,100,000,000 in scientific notation. Ex: Write the number 0.00032 in scientific notation. 5,100,000,000. 0 9 digits 0. 0 0 0 3 2 4 digits Decimal notation

11. 11 Ex 0.000043

12. If a is a real number and n > 2 is an integer, then Rational Exponents, the nth radical of a n is called the index of the radical a is called the radicand

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14. If a is a real number and m and n are integers containing no common factors with n > 2, then

15. 15 Radicals x Radicand ( المج | ور ) Radical n ( الرتبة ) Index ( الج \ ور )

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17. Properties of Rational Exponents If m and n represent rational numbers and a and b are positive real number, then

18. 18 Ex: Simplify

19. 19 Square Roots

20. 20 Square Roots continued ( الج|ر الرئيسي )

21. 21 Expressions in the form

22. 22 Simplifying

23. 23 Higher Order Roots Having an Index Larger than 2

24. 24 Ex:

25. 25 Ex:

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27. 27 Ex: Simplify:

28. 28 Multiplying Radicals

29. 29 Ex:

30. 30 Dividing Radicals

31. 31 Simplifying A Radical: For a radical to be simplified, the radicand cannot contain any factors that are perfect roots (i.e. exponents are evenly divisible by the index). To simplify the radical we do the following :  Factor the radicand into prime factors using exponential notation (or, express the radicand as a product of factors in which one factor is the largest perfect nth power possible).

32. 32 Use the product rule and the laws of exponents to rewrite the radical as a product of two radicals such that: a. First radicand: contains factors that are perfect roots (i.e. exponents are evenly divisible by the index). b. Second radicand: contains factors are not perfect roots (the indices are smaller than the index). Extract the perfect root from the first radicand.

33. 33 Like Radicals: Addition/Subtraction Ex: Simplify

34. 34 Ex:

35. 35 Ex: Simplify

36. 36 Rationalizing Denominators For an expression containing a radical to be in simplest form, a radical cannot appear in the denominator The process of removing a radical from the denominator or the numerator of a fraction is called rationalizing the denominator. ( انطاق الج | ور )

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39. 39 Simplify Multiply by the conjugate Ex: Rationalize the denominator of the following expressions:

40. Ex: Simplify each expression. Express the answer so only positive exponents occur.

43. 43 You’re shining!