Properties of Positive Integer Exponents, m > 0, n > 0

• for example,

• for example,

• for example,

,nmnm aaa

,)( mnnm aa

,)( mmm baab

.222 532

.2)2( 623

.32)32( 222

More Properties of Positive Integer Exponents, m > 0, n > 0

• for example,

• for example,

• for example,

,m

mm

b

a

b

a

,, nmaa

a nmn

m

.2

6

2

62

22

.22

2 32

5

,,1

nmaa

anmm

n

.2

1

2

235

2

Properties of Integer Exponents, a 0

• for example,

• for example,

• for example,

,10 a .12

10

,1

mm

aa .

2

12

33

,1m

m

aa .

2

12

33

Scientific Notation• A number is written in scientific notation if it is of the form

where 1 a < 10, and m is some integer.

• Scientific notation is useful for writing very large or very small numbers since fewer zeros are required in the representation of the number.

• Example. One light-year is about 6 trillion miles, which is 6,000,000,000,000 miles using decimal notation. This may be written as using scientific notation.

,10ma

12106

More about Scientific Notation• If the result of a measurement or calculation is written in

scientific notation as where 1 a < 10, and m is some integer, then the number of digits of a are taken as the significant digits of the result.

• If we write a number in scientific notation with fewer significant digits than the original number presented, we must round the last significant digit according to the rule: add 1 to last significant digit if digit following it in the original number is 5, 6, 7, 8, or 9

leave the last significant digit alone if digit following it in the original number is 0, 1, 2, 3, or 4

• Example. The speed of light is 186,282 miles per second. This is 1.863 105 to four significant digits.

,10ma

Summary of Integer Exponents; We discussed

• Six properties for positive integer exponents

• Three properties for general integer exponents

• Scientific notation

• Significant digits

Rational Exponents and Radicals• If n is a natural number and n is odd, b1/n can be defined to be

the unique nth root of b.

• If n is a natural number and n is even, b1/n can be defined to be the unique positive nth root of b. In this case, we say that b1/n is the principal nth root of b.

• Now, we define bm/n for an integer m, a natural number n, and a real number b, by

where b must be positive when n is even. With this definition, all the rules of exponents continue to hold when the exponents are rational numbers.

nmmnnm bbb /1/1/ )()(

Examples for Rational Exponents

• Simplify 274/3. We have

• Simplify We have

• Simplify . We have

• Simplify 163/4. We have Note that 161/4 is taken to be the principal root, which is 2.

.813)27(27 443/13/4

.)(4

41222/1

a

bbaba .)( 222/1 ba

12

6/5

3/23/1

z

yx12

6/5

3/23/1

z

yx .10

84

z

yx

.82)16(16 334/14/3

Radicals

• The principal nth root of a real number b was discussed previously. An alternative representation of this root is available using a radical symbol, . We summarize as follows.

with these restrictions:

if n is even and b < 0, is not a real number, if n is even and b 0, is the nonnegative number a satisfying an = b.

• Warning.

baabb nnn where,/1

n b n b

.24 ,24

Radicals and Exponents

• We have that

• Example.

• Problem. Change from rational exponent form to radical form, Assume that x and y are positive real numbers.

Solution.

.)()(

and ,)(/1/

/1/

mnmnnm

n mnmnm

bbb

bbb

???)8()8(

8)8(82323/1

3 23/123/2

.)( 2/5yx

.)()(552/5 yxyxyx

Properties of Radicals

• for example,

• for example,

• for example,

,m

nn m bb .882

33 2

,nnn abba .3694

,nn

n

b

a

b

a .

27

8

27

83

3

3

More about Radicals

• For example,

• For example,

• Simplify

. odd, is If aan n n .2)2(3 3

.|| even, is If aan n n .2 |2| )2( 2

.55555

3335

44444

444

.5

3

5

3

5

3

)5(5

)3(3

55555

3335 5

5 5

55

5

54

4

544444

444

Rationalizing Denominators

• A fraction is sometimes considered simplified if its denominator is free of radicals. The process by which this is accomplished is called rationalizing the denominator.

• In this connection, a useful formula is:

• Example. Rationalize the denominator:

.nmnmnm

.25

4

253

4

25

254

25

25

25

4

25

4

Examples for Simplifying Expressions Involving Radicals

• .

•

.

8/72/14/74/32/1 xxxxxxxxxx

x

x

x

xx

xx

xxxxxxxxx

x

22

12/12/12/12/12/12/1

2

)1(1212

21

Rational Expressions and Radicals; We discussed

• nth roots and the principal nth root

• rational exponents

• radicals

• converting from rational exponent form to radical form and vice versa

• properties of radicals

• rationalizing the denominator