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

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4 Laws of Exponents LawExample

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6 Ex: Simplify the following expression

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8 More Examples

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 < 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 Ex : Write the number 5,100,000,000 in scientific notation. Ex: Write the number in scientific notation. 5,100,000, digits digits Decimal notation

11 Ex

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

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

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

18 Ex: Simplify

19 Square Roots

20 Square Roots continued ( الج|ر الرئيسي )

21 Expressions in the form

22 Simplifying

23 Higher Order Roots Having an Index Larger than 2

24 Ex:

25 Ex:

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27 Ex: Simplify:

28 Multiplying Radicals

29 Ex:

30 Dividing Radicals

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 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 Like Radicals: Addition/Subtraction Ex: Simplify

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35 Ex: Simplify

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

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

43 You’re shining!