[PDF] Lecture 2: Section 12: Exponents and Radicals Positive Integer

1287_6Lecture2,ExponentsandRadicals.pdf

Lecture 2: Section 1.2: Exponents and Radicals

Positive Integer Exponents

Ifais any real number andnis any natural number (positive integer), thenth power ofais dened as a n=a
a
a




a|{z} n factors

ExampleEvaluate the following:

(2)3;24;(1)5(1)4;13  3;03 the numberais called thebaseandnis called theexponent.

Integer Exponents

To make the above denition work for exponents which are 0 or negative integers, we must restrict the

possibilities for the base to non-zero real numbers.

Zero and Negative Exponents

Ifa6= 0 is any non-zero real number andnis a positive integer, then we dene a

0= 1 andan=1a

n:

ExampleEvaluate the following

(2)0;23;(3)213  2: 1 Laws of Exponents for Integer ExponentsThe following algebraic rules apply to exponents:

Rule Example Description

1.aman=am+n35310= 315When multiplying two powers of the same number,

add the powers. 2. ama n=amn353

10= 3510= 35When dividing two powers of the same number,

subtract the exponents.

3. (am)n=amn(35)10= 350When raising a power of a number to a new power,

multiply the powers.

4. (ab)n=anbn(3
5)10= 310
510A product of two numbers raised to a given power,

is the same as the product of the factors raised to the given power. 5. ab
n=anb n35

10=3105

10Raising a quotient of two numbers to a given power

is the same as raising the numerator and the denominator to the given power. 6. ab
n=bna n35
10=5103

10To raise a fraction to a negative power,

you can ip the fraction and change the sign of the exponent. The proofs of these rules are not very dicult and can be found in the textbook. For example to prove the rst law whenmandnare positive integers, we see that a man=a
a
a




a|{z} a
a
a
a




a|{z} =a
a
a
a
a
a




a|{z} =am+n n factors m factorsm+nfactors ExampleSimplify the following expressions (herexcan have any real number value andzandycan have only non-zero values): (a)x3x2+x4x1. (b) (x3)2x4x 5 (c) xyz

3x2z2zy

. (d) yz
3x2z2zy . 2 Scientic NotationA positive numberxis said to be written in scientic notation if it is expressed as x=a10n where 1a <10 andnis an integer. ExampleWrite the following numbers in scientic notation: :000004;100:12 ExampleWrite the following numbers in decimal form:

2:31103;1:31103

Radical Powers

Square RootsFor any real numbera0, we denea1=2in the following way a

1=2=bmeansb2=aandb0:

We also use the notation

pato denotea1=2and we callpa, the square root ofa.

Example

p9 = 3 because 3

2= 9 and 3>0.p16 = 4 because 4

2= 16 and 4>0.

For a real numbera, and a positive integern, we denea1=nin a similar way a

1=n=bmeansbn=a:

Ifnis even, we must havea0 andb0 in this denition.

We also use the notation

npa=a1=nand refer tonpaas thenth root ofa.

ExampleEvaluate the following:

3p8;4p16

5p32 3 Properties ofnth RootsThe algebraic rules of exponents given above for integer exponents work fornth roots. Since we have not denedaxwherexis a general rational number yet, we rst generalize rules 4, 5 and 3 above.

Rule Example

(similar to rule 4 above) 1. npab=npa npbor (ab)1=n=a1=nb1=n(278)1=3= (27)1=3(8)1=3= 32 = 6 (similar to rule 5 above) 2. npa b =npa n pb orab

1=n=a1=nb

1=n278

1=3=(27)1=3(8)

1=3=32

(similar to rule 3 above) 3. mpn pa=mnpaor (a1=n)1=m=a1=(mn)3p2 p64 =

3p8 = 2 =

6p64 4. npa n=aifnis odd.3p(3)3=3 5. npa n=jajifnis even.2p(3)2= 3

ExampleSimplify the following:

3 p27x3y6;p500 + p18

Rational Exponents

For any rational exponent (fractional exponent)m=n, wheremandnare integers,n >0 and the fraction is in lowest terms, we dene a m=n= (npa)mor equivalentlyam=n= (npa m):

Ifnis even, then we require thata0.

It is not too dicult to show that the laws of exponents hold for rational exponents. ifx=m1n

1andy=m2n

2are rational numbers as above, we can show that

1.axay=ax+y, (wherea0 ifn1orn2are even.)

2. axa y=axy,a6= 0, (wherea0 ifn1orn2are even.) 3. ( ax)y=axy, (wherea0 ifn1orn2are even.) 4. ( ab)x=axbx, (wherea;b0 ifn1is even.) 4 5. ab
x=axb x,b6= 0. (wherea;b0 ifn1is even.) In Calculus II, we will deneaxmore generally whenxis a real number anda >0. The above laws of

exponents will continue to hold for for these exponents. Note that the laws for integers andnth roots

are special cases of the above laws of exponents.

ExampleSimplify the following:



2x1=4y

1=5 5y2x  ;qx 3px 5