1 2 Objectives Integer Exponents Rules for Working

1. 2 Objectives ► Integer Exponents ► Rules for Working with Exponents ► Scientific Notation ► Radicals ► Rational Exponents ► Rationalizing the Denominator 1

Example 1 – Exponential Notation (a) (b) (– 3)4 = (– 3) = 81 (c) – 34 = –(3 3 3 3) = – 81 2

Integer Exponents A product of identical numbers is usually written in exponential notation. For example, 5 5 5 is written as 5 3. 3

Example 2 – Zero and Negative Exponents (a) =1 (b) (c) 4

Integer Exponents 5

Rules for Working with Exponents Familiarity with the following rules is essential for our work with exponents and bases. In the table the bases a and b are real numbers, and the exponents m and n are integers. 6

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Example 4 – Simplifying Expressions with Exponents Simplify: (2 a 3 b 2)(3 ab 4)3 Solution: (2 a 3 b 2)(3 ab 4)3 = (2 a 3 b 2)[33 a 3(b 4)3] Law 4: (ab)n = anbn = (2 a 3 b 2)(27 a 3 b 12) Law 3: (am)n = amn = (2)(27)a 3 a 3 b 2 b 12 Group factors with the same base = 54 a 6 b 14 Law 1: ambn = am + n 8

Example 4 Simplify Laws 5 and 4 Law 3 Group factors with the same base Laws 1 and 2 9

Rules for Working with Exponents Two additional laws that are useful in simplifying expressions with negative exponents. 10

Example 5 – Simplifying Expressions with Negative Exponents Eliminate negative exponents and simplify each expression. (a) (b) 11

Example 5 – Solution (a) We use Law 7, which allows us to move a number raised to a power from the numerator to the denominator (or vice versa) by changing the sign of the exponent. Law 7 Law 1 12

Example 5 – Solution cont’d (b) We use Law 6, which allows us to change the sign of the exponent of a fraction by inverting the fraction. Law 6 Laws 5 and 4 13

Scientific Notation For instance, when we state that the distance to the star Proxima Centauri is 4 1013 km, the positive exponent 13 indicates that the decimal point should be moved 13 places to the right: 14

Scientific Notation When we state that the mass of a hydrogen atom is 1. 66 10– 24 g, the exponent – 24 indicates that the decimal point should be moved 24 places to the left: 15

Radicals The symbol means “the positive square root of. ” Thus = b means b 2 = a ; b 0; a 0 16

Radicals 17

Example 8 – Simplifying Expressions Involving nth Roots (a) Factor out the largest cube Property 1: Property 4: (b) Property 1: Property 5, Property 5: 18

Example 9 – Combining Radicals (a) Factor out the largest squares Property 1: Distributive property (b) If b > 0, then Property 1: Property 5, b > 0 Distributive property 19

Rational Exponents A rational exponent or a fractional exponent such as a 1/3, is represented by a radical. To give meaning to the symbol a 1/n in a way that is consistent with the Laws of Exponents, we would have to have (a 1/n)n = a(1/n)n = a 1 = a The nth root a 1/n = 20

Example 11 – Using the Laws of Exponents with Rational Exponents (a) a 1/3 a 7/3 = a 8/3 Law 1: ambn = am +n (b) Law 1, Law 2: = a 2/5 + 7/5 – 3/5 = a 6/5 (c) (2 a 3 b 4)3/2 = 23/2(a 3)3/2(b 4)3/2 =( =2 )3 a 3(3/2)b 4(3/2) Law 4: (abc)n = anbncn Law 3: (am)n = amn a 9/2 b 6 21

Example 11 – Using the Laws of Exponents with Rational Exponents (d) Laws 5, 4, and 7 Law 3 Law 1, Law 2 22

Rationalizing the Denominator It is often useful to eliminate the radical in a denominator by multiplying both numerator and denominator by an appropriate expression. This procedure is called rationalizing the denominator. If the denominator is of the form , we multiply numerator and denominator by. In doing this we multiply the given quantity by 1, so we do not change its value. For instance, 23

Example 13 – Rationalizing Denominators (a) (b) (c) 24