1.2 Objectives Integer Exponents Rules for Working with Exponents Scientific Notation Radicals Rational Exponents Rationalizing the Denominator

Integer Exponents A product of identical numbers is usually written in exponential notation. For example, 5  5  5 is written as 53. In general, we have the following definition.

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

Integer Exponents

Example 2 – Zero and Negative Exponents (b)

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.

Example 4 – Simplifying Expressions with Exponents (a) (2a3b2)(3ab4)3 (b) Solution: (a) (2a3b2)(3ab4)3 = (2a3b2)[33a3(b4)3] = (2a3b2)(27a3b12) = (2)(27)a3a3b2b12 Law 4: (ab)n = anbn Law 3: (am)n = amn Group factors with the same base

Example 4 – Solution = 54a6b14 (b) cont’d Law 1: ambn = am + n Laws 5 and 4 Law 3 Group factors with the same base Laws 1 and 2

Rules for Working with Exponents We now give two additional laws that are useful in simplifying expressions with negative exponents.

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

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

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

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:

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:

Example 6 – Changing from Decimal to Scientific Notation Write each number in scientific notation. (a) 56,920 (b) 0.000093 Solution: (a) 56,920 = 5.692  104 (b) 0.000093 = 9.3  10–5

Radicals We know what 2n means whenever n is an integer. To give meaning to a power, such as 24/5, whose exponent is a rational number, we need to discuss radicals. The symbol means “the positive square root of.” Thus = b means b2 = a and b  0 Since a = b2  0, the symbol makes sense only when a  0. For instance, = 3 because 32 = 9 and 3  0

Radicals Square roots are special cases of nth roots. The nth root of x is the number that, when raised to the nth power, gives x.

Radicals

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

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

Rational Exponents To define what is meant by a rational exponent or, equivalently, a fractional exponent such as a1/3, we need to use radicals. To give meaning to the symbol a1/n in a way that is consistent with the Laws of Exponents, we would have to have (a1/n)n = a(1/n)n = a1 = a So by the definition of nth root, a1/n =

Rational Exponents In general, we define rational exponents as follows.

Example 11 – Using the Laws of Exponents with Rational Exponents (a) a1/3a7/3 = a8/3 (b) (2a3b4)3/2 = 23/2(a3)3/2(b4)3/2 = ( )3a3(3/2)b4(3/2) = 2 a9/2b6 Law 1: ambn = am +n Law 1, Law 2: Law 4: (abc)n = anbncn Law 3: (am)n = amn

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,

Example 13 – Rationalizing Denominators (b)