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Exponents and Radicals A product of identical numbers is usually written in exponential notation. For example, 5 . 5 . 5 = 53 . In general, we have the following definition. !Exponential Notation: If a is any real number and n is a positive integer, then the n nth ower of a is The number a is called the base and the number n is called the ex '----------. a . Example 1: Exponential notation \ '- 0 \ . '-O- \ - d- d \ - Notice a pattern from example 1 part b? What did you do with the exponents? Zeros and Negative EXp'onents: anda -n If a;j:. 0 is an real number and n is a ositive integer, then aO = 1 =~. a Example 2: Zero and negative exponents. a) . \ -- -d.- 1 .: Laws of Exponents: Description Law ~ti)Q l. aman - a n+ n To multiply ~owers of the same number, add t e exponents To divide two powers of the same number, subtract the exponents. m 2. a -=a an 3. (am 4. (aby ==a nbn To raise a product to a power, raise each factor to the power. J = ~: 6(:r =(!J To raise a quotient to a power, raise both numerator and denominator to the power. m=n t ==a m·n To raise a power to a new power, multiply the exponents. 5. (~ 7. a -n b-m To raise a fraction to a negative power, invert the fraction and change the sign of the exponent. bm -an - To move a number raised to a power form numerator to denominator or from denominator to numerator, change the sign of the exponent. Example 3: Simplifying expressions with exponents a) (2a3b2)(3ab4)3 = (J~O:~ <> ') l3 3 - (,)03'0" Sb) 6st-4 2s-2t2 ~ ClocI)') &~\5~) L{ CL~ \01 '-I d V ~ ..~ C} t ~t:-J LOUJJ 3 Rational Exponents: (\3 ~ I _\ ~ ~ WOJV\/-\- GtJuld ~ Yl~ ~ ~ efinition of rational exponents:'--__ --------~......, or any rational exponent mln in lowest terms, wherem and n are integers and n > 0, we define a'% = (~t or equivalently a% = ~. [f n is even, then we re uire a ~ 0 . Example 4: Rational Exponents li a) 4 \ \ Rationalizing the Denominator It is often useful to eliminate the radical in a denominator by multiplying both the numerator and denominator by an appropriate expression. 2 a) - J3 • 1 b) ifI2~ 3 X n~ • ~ - ~ c) V:2 ~ ~ ~o1 ~ ~ ..., -- -: ~ ~·M~~ L~ K )(d~ ~ \ • X$ ~o!l Zj&1 Sl ~ Q::> ~