mat 1033-l8-sat-sections 7-1-2-3-4

8/17/2019 mat 1033-l8-sat-sections 7-1-2-3-4

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7.1 – Radicals

Radical Expressions

Finding a root of a number is the inverse operation of raisinga number to a power.

This symbol is the radical or the radical sign

n a

indexradical sign

radicand

The expression under the radical sign is the radicand.

The index defines the root to be taen.


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Radical Expressions

The symbol represents the negative root of a number.

The above symbol represents the positive or principal

root of a number.

−

7.1 – Radicals


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!"uare Roots

#f a is a positive number$ then

a is the positive s"uare root of a and

1%% =

a− is the negative s"uare root of a.

& s"uare root of any positive number has two roots – one is positive and the other is negative.

Examples'

1%

()

*+

=

)

7

%.,1 = %.+

-− = −

+− = non/real 0

=

, x * x

7.1 – Radicals


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Rdicals

ube Roots

- (7 =

& cube root of any positive number is positive.

Examples'

-)

*

-1()

*=

- ,− = (

−

& cube root of any negative number is negative.

- a

=

- - x x =

- 1( x* x

7.1 – Radicals


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nth Roots

&n nth

root of any number a is a number whose nth

power is a.Examples'

(

*

,1 =

-

* 1 =

) -(− = (−

*

- =

,1

*( = 1

( ) )

(− = -(−

7.1 – Radicals


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nth Roots

* 1− =

&n nth

root of any number a is a number whose nth

power is a.Examples'

1−) 1− =

2on/real number

. 1− = 2on/real number

- (7− = -−

7.1 – Radicals


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7.( – Rational Exponents

The value of the numerator represents the power of theradicand.

Examples'

'nm

aof Definition

The value of the denominator represents the index or root of

the expression.

n ma or

mn a

-1

(7()(1

() -) -

(7

7(

1(

x

-*(

-

* *

7 (1(

x

,


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3ore Examples'

'nm

aof Definition n ma or m

n a

-(

-(

(7

1-(

(7

1

- (

- (

(7

1

+

1

-

-

7(+

1

-(

-(

(7

1-(

(7

1

(-

(-

(7

1

+

1 (

(

-

1

or


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Examples'

'nm

aof Definition

n m

a

1

m

n a

1

(1

()1(1()

()1

)1

-(

1

x-

(

x - (

1

x (

-

1

x

n

m

a

1or or

or


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7.( – Rational Exponents4se the properties of exponents to simplify each expression

-)-* x x ⋅

-+ x - x

1%1

)-

x1%1

)-

x

x1%

11%

x 1%)

x

*(

- x* (

,1 x

(1

- x

-)-* x

(1

x

- (1( x x ⋅ 1(

,1(

1

x 1(+

x *-

x-(

1(1

x x


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*% =

Examples'

* 1%× =

#f and are real numbers$ then a ba b a b× = ×

5roduct Rule for !"uare Roots

( 1%

7 7) = 7 () -× = 7 ) -× = -) -

7.- – !implifying Rational Expressions

=

17

1 x = x x1

1 x x,

*

=

- 171. x

=

- (1)(, x x - ()

(( x x

=

1%*

=

-()7


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1)

-=

+%

(

=

a#f and are real numbers and %$ then

b

aa b b

b

≠ =

- )

-

×=

- )

-

×= )

+ 1%

(

×=

+ ( )

(

× ×=

+ ( )

(

× ×= - )



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11 x =

Examples'

77

()

y=

,

(7

x=

7

()

y y×=

-7

)

y y

1% x x× = ) x x

*1, x = *+ ( x× = (- ( x

,

+ -

x

×=

*

- -

x,(7

x=



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-

,, =

Examples'

-,1

,=

-1%

(7=

-

-

,1

,=

- (7 -

(

×=

-

, 11× = -

( 11

- 1%

-

-

-

1%

(7=

-- -

(


- - 7(7m n = - - - m n n =

( --mn n


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ne 8ig Final Example

1( * 1,) * x y z =

1% ( * 1) -) -( ( x x y z z × =

( - ( * -)

( ( x z x y z



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) - x x+ =

Review and Examples'

11 + 11+ =

, x

1) 11

1( 7 y y− = ) y

7 - 7− = ( 7−

7.* – &dding$ !ubtracting$ 3ultiplying Radical

Expressions


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(7 7)+ =

!implifying Radicals 5rior to &dding or !ubtracting

- (% 7 *)− =

+ - () -× + × =

- * ) 7 + )× − × =

- - ) -+ = , -

- ( ) 7 - )× − × =

) (1 )− = 1) )−

- *, * - +− − − = 1 - * - -− × − − =

* - * - -− − − = - , -−


Expressions


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) (× =

7 7× =

1% ( x x× =

#f and are real numbers$ then a ba b a b× = ×

1%

*+ = 7

-× = 1, = + (× = - (

((% x = (* ) x× = ( ) x


Expressions


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( )7 7 -− = 7 7 7 -× − × = *+ (1− =

( )) - ) x x − =

( ) ( )) - x x+ − =

7 (1−

() - () x x− = ) - ) x x− × =

) 1) x x−

(- ) 1) x x x− + − =

(- ) 1) x x x− + −


Expressions


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( ) ( )- - + − =

( ) (

) * x + =

+ - - -− + − = - -− =

--−

( ) ( )) * ) * x x+ + =

(() * ) * ) 1 x x x+ + + =

) , ) 1 x x+ +


Expressions

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