P.2 2 INTEGER AND RA INTEGER AND TIONAL

P.

P.2 2 INTEGER AND RATIONAL INTEGER AND RATIONAL NUMBER EXPONENTS

NUMBER EXPONENTS

(ﺔﻴﺒﺴﻨﻟاو ﺔﺤﻴﺤﺼﻟا ﺲﺳﻷا)

( ﻴﺒ و ﻴ ﺲ )

Objectives: j

In this lecture, you learn about:

1.

Properties of Exponents

2.

Scientific Notation

3.

Rational Exponents and Radicals

4

Simplify Radical Expression

4.

Simplify Radical Expression

Definition of Natural Number Exponents

n

If a is a real number and n is a natural number, then ,

a n = ⋅ ⋅a a … ⋅ a

n times

h i h b ( ﺎ ﻷا) d i h ( ﻷا)

where a is the base (سﺎﺳﻷا)and n is the exponent (سﻷا).

E

4

= ⋅ ⋅ 4 4 4

Ex:

a

b. .

− 2

= − ⋅ ⋅ ⋅ ⋅ ⋅ (2 2 2 2 2 2) = − 64

c

( ) ²

( 2) ( 2) ( 2) ( 2) ( 2) ( 2) 64

(base is -2) c.

( ) ^{− 2} = − ⋅ − ⋅ − ⋅ − ⋅ − ⋅ − ( 2) ( 2) ( 2) ( 2) ( 2) ( 2) = 64

to the power the po

: wer of

n th

b n n b

b or

2 3

: b squa e r b

3

: cub e

b b

For any nonzero real number , b b

= 1, b ≠ 0

Def. of b ⁰

Ex: a 6

1 _{b 6}

Ex: a. 6 = 1 _{b . 6 −}

_{= − (6 )}

_{= − 1}

(

)

. 5 1

c x ( + ) =

Definition of b ^{− n}

1 1-

If b 0 and n is a natural number, then b ⁿ _n and _n bⁿ

b b

≠ − = =

Definition of b ⁿ

2 3

2t

− −

(This property applies to factors only( not to terms))

Ex. ^{2 3 5}

1 5 2 3

. 2 2

y x t

a

t

= y x

( )

2 3

5 2 3

1 5

. 2

2

y x

b t y x

t

− −

− −

−

+

= ( + )

2 t

Restriction Agreement

The expressions ,0 (wherenisa negati a

0 0ⁿ veinteger), and are all undefined.0

Properties of Exponents

If m, n, and p are integers and a, and b are real numbers, then Product b b^{m n} = b^{m n+}

1

m m

Quotient Power

, 1 ,

m m

m n

n n n m

a a if m n o r a if n m

a a a

− −

= > = >

( )

Power

( )

( )

( )

( ) ^b

^{= b}

^, ( ^{a b}

)

^{= a}

^b

To simplify an expression involving exponents, write the expression in a from in which:

1. Each base occurs at most once 2. No powers of powers

3. No negative exponents occur

Ex: Simplify

(

)(

)

. 3 6

a − ab − a b

Ex: Simplify

Sol ^{= − =} ( )( ) ( )( ) ^{3 3 − 6 a b 6 a b}

on the like bases

3 0 3

18 a b 18 a

= =

( )

2 2

. 2 3

b − ab − a b

Sol ^{2 ab}

( ) ³

( ) ^a

^b

= − −

( )

2 6 3

( )

Sol

( )

2 6 3

2 ab 27 a b

= − − ^{= − − 2 27 a b} ( )

= 54a b

2 2

2x y ⁻

⎛ ⎞

2 3

. 2 6 c x y

x y⁻

⎛ − ⎞

⎜ ⎟

⎝ ⎠

Sol Sol

( ) ²

2 2

3 3 1

x ^{− − −}

−

⎛ − ⎞

=⎜ ⎟

⎝ ⎠

•Divide the coefficients

•Divide the variables by subtracting the exponents h lik b

4 2

x ⁻

⎛ − ⎞

= ⎜ ⎟

3y3 1

⎜ ⎟

⎝ ⎠ on the like bases

3y2

= ⎜⎝ ⎟⎠

2 2

⎛ 3 y ⎞

= ⎜ ⎟

Use the negative property of exponents

x4

= −⎜ ⎟

⎝ ⎠

9y4 Use the power property of exponents

8

y

= x ^{p p p y p}

2 2 1 3 2

( a b⁻ )⁻ c ⁻

⎛ − ⎞

2 2

( )

. 2

a b c d a c⁻

⎛ ⎞

⎜ ⎟

⎝ ⎠

8 4 10

4a

= b c

2 2 3 2 2 2

2

a b c a c

− −

−

⎛ − ⎞

= ⎜ ⎟

⎝ ⎠

2 5 2

2 4

b c a

⎛ − ⎞−

= ⎜ ⎟

⎝ ⎠

4 2 2 5

2a b c

⎛ − ⎞

= ⎜ ⎟

⎝ ⎠ b c

⎝ ⎠ ⎝ b c ⎠

Scientific Notation for Writing Numbers ﺔﻴﻤﻠﻌﻟا ﺔﻐﻴﺼﻟا 1 0 ⁿ

± a ×

1 ≤ 1 0

integer

1 ≤ a < 1 0

Ex Write each of the following numbers in scientific notation Ex Write each of the following numbers in scientific notation

1)3478.28 = 3.4828 10× ³

3 digits

2)0.0000078 = 7.8 10× ⁻⁶

6digits

3) 345700000− = −3 457 10× ⁸ 3) 345700000 = −3.457 10×

Ex Write in decimal notation

1) -3.253 10⁷ = -32530000.

7 decimal places

×

2) 4.25 10^-5 = 0.0000425

5 decimal places

×

Ex Write the following expression in scientific notation

11 18

(3.2 10 )(2.7 10× ⁻ ×

5

( )(

1.2 10× ⁻

11 18

Sol

11 18

11 18 ( 5) 5

11 18 5

(3.2 10 )(2.7 10 ) (3.2)(2.7) 1.2 10 1.2 10

7 2 10

− − + − −

−

− + +

× ×

= ×

×

11 18 5

7.2 10 ^{+ +}

= ×

Rational Exponents and Radicals روﺬﺠﻟاو ﺔﻴﺒﺴﻨﻟا ﺲﺳﻷا Definition of 1

1. If n is an even positive integer and , then is the

1

bn 1

bn

b≥0

nonnegative real number such that⎛ ⎞b^{1 nn} b

⎜ ⎟ =

⎝ ⎠

2. If n is an odd positive integer, then is the real number such that

1

bn

⎛ ⎞1 n_n

b b

⎛ ⎞ =

⎜ ⎟⎝ ⎠

1/2 2

a. 25 =5 since 5 = 25

Ex

( )

( )

. 27 3 since 3 27

b − = − − = −

( )¹

1 / 2 2

. 4 4 2

c − = − ( ) = −

. ( 4) is noe defined in real numbers1/ 2

d − e. ( 32) = 2 since 2− ¹⁵ − ( )− ⁵ = −32

Def. of b^mn

Def. of

For all positive integers n, all integers m, such that is in simplest form and all real numbers b such that b¹ⁿ is a real

bn

m n

p

number, b

( )

^{( )}

/ m

m n n m n

b = b = b

Ex: Evaluate a) 125²³

It can be evaluated in either of the following ways

( )

^{( )}

23 2 3 13

125 = 125 = 15625 = 25

( )

g y

( )

^{( )}

23 13 2

125 = 125 = 5 = 25 ) 81 0.75

b ⁻

1 0.75

⎛ ⎞

= ⎜ ⎟ ⎛ 1 ⎞ ³⁴

1 3

1 4

⎛ ⎛ ⎞ ⎞

⎜ ⎜ ⎟ ⎟

1 3 1

⎛ ⎞ ) 81

b = ⎜ 81 ⎟

⎝ ⎠ 1

81

⎛ ⎞

= ⎜⎝ ⎟⎠ = ⎜⎜ ⎜⎝ 81 ⎟⎠ ⎟⎟

1 1

3 27

= ⎛ ⎞⎜ ⎟ =

⎝ ⎠

Def. ^{n b}

If n is a positive integer and b is areal number such that is a real number then

1

b n 1

n b = b n

radical index

is a real number, then ^b = ^b

n

b

radical symbol

radicand

Common Roots

the sq ar root of

: e

b b

radicand

3

the root of the root o

squ

f ar

: :

e cube b

b

b b

the root of , 3 :

n th

b fo

n r b

b >

Def. of ⁿ b^m Def. of

For all positive integers n, all integers m, and all real numbers b such that ⁿb is a real number,

b

,

( )

n m n m n

b = b = b

2 / 3

) 125 ³ ₁₂ ² ₂ Ex: Find the value of

2 / 3

) 125

a ₌ ³ ₁₂₅² _{= 25}

alternatively, it can be evaluated as follows

( )

^{( )}

= 125 = 5 = 25

3/ 2 3

⎛ ⎞

2 / 3

125 Easer to evaluate

9 3 27

⎛ ⎞

4 3/ 2

)

⎛ ⎞−

Def. of ⁿ bⁿ Def. of

• If n is an even natural number and b is a real number, then

n n

b b

b

b

= b

• If n is an odd natural number and b is a real number, then

n n

b = b

Ex

4 4 4 4 ³ x^{3 3} ( x)³ x

4

( 12) −

= − 12 12 =

5 ( 12)5 12

4

16 x

=

(2 ) x

= 2 x

5

32

(2 )

2

3 ( )

x x x

− = − = −

5 ( 12)− 5 = −12 ⁵

32 x

=

(2 ) x

= 2 x

Ex

Ex:: Simplify each exponential expression.

6 3 0.4 13

. 32x y a xy

⎛ ⎞−

⎜ ⎟

⎝ ⎠

Sol

y4 10 2 5/

⎛ y ⎞

⎜ ⎟

10 2

5 y

⎛ ⎞

⎜ ⎟

2 2

⎛ y ⎞

4 3 1/ 6

3 −

⎛ ⎞

4 2

y

= x 32 5

y x

⎛ ⎞

= ⎜ ⎟

⎝ ⎠

5 32 5

y

= ⎜⎜⎝ x ⎟⎟⎠ 2 y

x

⎛ ⎞

= ⎜ ⎟

⎝ ⎠

3 4 3

4 9

. x y z , 0, 0

b y z

xy z

−

−

⎛ ⎞

> >

⎜ ⎟

⎜ ⎟

⎝ ⎠

Sol

x1/ 9y

1/ 3 2 3 1/ 6

x y z^{− −}

⎛ ⎞

⎜ ⎟

12 1/ 6

z ⁻

⎛ ⎞ ⎛ x^{2 / 3 6}y ⎞^{1/ 6} _{x y}

=

4 9

y

−

⎛ ⎞

= ⎜ ⎟

⎝ ⎠ ^{2 / 3 6}

⎛ z ⎞

= ⎜ ⎟ ⎛ x y ⎞

= ⎜ ⎟

Properties of Radicals

If m and n are natural numbers and a and b are positive real numbers, then

Product

a ⋅

b =

ab

8 ⋅

27 =

216 = 6

Quotient

, 0

n

a a b

b = b ≠ ⁸ ₂ ^{= 8} ₂ ^{= 4 = 2}

Q ,

n

b b ^{2 2}

3 4

26

26

Index

a =

a

26 =

26

An expression involving radicals is said to be in simplest form if it meets all of the following criteria.

1. The radicand contains only powers less than the index.

24 = 4 ⋅ = 6 2

⋅ = 6 2 6

2. The index of the radical is as small as possible.

3 Th d i t h b ti li d ( di l

p

3

125 =

125 =

(5)

= 5

= 5

= 5

3. The denominators have been rationalized (no radicals appear in the denominator).

1 = 1 ⋅ 5 = 5 5 = 5 ⋅ 5 = 5

4. No fractions appear under the radicals sign.

1 = 1 = 1 ⋅ 5 = 5

Ex

Ex Simplify

a) 75

Sol

7 5 = 5 2 ⋅ 3 = 5 3

2 3

b ) 2 4 x y

Sol

2 3 2 2 2

24 x y = 2 x y ⋅ 6 y = 2 xy 6 y

Addition and Subtraction of Radicals

We add and subtract like radicals (the same radicand and the same index)e s e de )

Ex

Ex1010:: Perform the following operations 5

3 3 3 3

a ) + − = 3 3

2 2

3 3

3 2

)

b xy − xy =

c ) − = 3 25 3 2 16 3⋅ − ⋅ ^{=15 3 − 8 3 = 7 3}

Ex Simplify

3 4 6

3 3

. 3 8 4 64

a x x y + y x y

Solution

3 2

3 3 1 3 3 3

3x 8x y y 4y 4 ( )x y

= ⋅ + ⋅

3 3 3 3 2 3

3 3 3 3 3 3 3 2 3

3 3 3

3 2x x y y 4y 4 ( )x y

= ⋅ +

3 2 3

3 (2 ) x xy y 4 (4 ) y x y

= ( y ) y + y ( ) y

2 3 2 3

6 x y y 16 x y y

= +

2 3

22x y y

=

3 4 3 7

. 3 54 2 16

b − x x + x

Sol

3 4 3 7 3 3 3 6

3 3 3 3 6 3

3 54 2 16 3 27 2 2 8 2

3 27 2 2 8 2

x x x x x x x x

x x x x x

− + = − ⋅ + ⋅

= − ⋅ + ⋅

3 3 3 3 3 3 6 3

3 2 3

3 27 2 2 8 2

3 3 2 2 2 2

3 (3 ) 2 2(2 ) 2

x x x x x

x x x x x

x x x x x

+

= − ⋅ + ⋅

3 + 3

2 3 2 3 2 3

3 (3 ) 2 2(2 ) 2

9 2 4 2 5 2

x x x x x

x x x x x x

= − +

= − + = −

3 3

2 40 3 135 c. 2 40 3 135− c

Solution

3 3 3 3 3 3

3 3 3 3 3 3

2 8 5 3 27 5 2 8 5 3 27 5

2 2 5 3 3 5

⋅ − ⋅ = ⋅ − ⋅

= ⋅ − ⋅

3 3 3 3 3 3

3 3

2 2 5 3 3 5

2(2) 5 3(3) 5

= ⋅ − ⋅

= −

3 3

3

2(2) 5 3(3) 5 4 5 9 5

5 5

=

= −

= 5 5

= −

Multiplication of Radicals

Multiplication of radicals is accomplished by using the distributive property.

Ex: Perform the following operations

2 8 2 2 a. 2( 8 + 2 2 )

a. ( + )

( )

2 8 + 2 2 2 ^{= 16 + 2(2)} = + =4 4 8 2 ) ( 3 + 5 )( 3 − 5 )

3 5 3

= − + 5 3 − 25 = −22

( ) ( )

3 3 3 5 5 3 5 5

= + − + + −

2

3) (3 5y − 4)2

( )

2 2

3 5y + 2(3 5 )( 4) ( 4)y − + − =9(5 ) 24 5y − y+16 ^{= 45y − 24 5y +16}

Rationalizing the Denominator مﺎﻘﻤﻟا قﺎﻄﻧإ

Rationalizing the denominator means writing the fraction with out radicals in the denominator.

م

Ex Rationalize the denominator.

) 1 1 a.)

2

Sol

1 2

= ⋅ 2

= 2

1

Sol

2 2 2

2

conjugate

2 3 )

b −

Sol

2 3

) 2 3

b +

2 3

2 3

2 3

2 3

= −

+ i −

Sol

( )

( ) ( )

2

2 2

2 3

2 3

= −

2 + 3 2 − 3

2 2 6 3

− 1 +

= − = ^{5 2 6}−−1 =− +^{5 2 6}

( ) ( )

) 1 c )

1

c x

Sol

3 3

3 2 3 2

1 = 1 i x

Sol

3 2

3 3

= x ^{3 x2}

= x

3 x 3 x 3 x2

) 1

3 3

x ^x

5 2

) 1 c 8

x

S l

5 2 3 5

4

4

1 1

2 x x

2

= i = =

Sol

5

8

2

2

2