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Transcript of 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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7.( – Rational Exponents
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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7.( – Rational Exponents
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%
(
×=
+ ( )
(
× ×=
+ ( )
(
× ×= - )
7.- – !implifying Rational Expressions
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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=
7.- – !implifying Rational Expressions
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-
,, =
Examples'
-,1
,=
-1%
(7=
-
-
,1
,=
- (7 -
(
×=
-
, 11× = -
( 11
- 1%
-
-
-
1%
(7=
-- -
(
7.- – !implifying Rational Expressions
- - 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
7.- – !implifying Rational Expressions
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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 - * - -− × − − =
* - * - -− − − = - , -−
7.* – &dding$ !ubtracting$ 3ultiplying Radical
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
7.* – &dding$ !ubtracting$ 3ultiplying Radical
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− + −
7.* – &dding$ !ubtracting$ 3ultiplying Radical
Expressions
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( ) ( )- - + − =
( ) (
) * x + =
+ - - -− + − = - -− =
--−
( ) ( )) * ) * x x+ + =
(() * ) * ) 1 x x x+ + + =
) , ) 1 x x+ +
7.* – &dding$ !ubtracting$ 3ultiplying Radical
Expressions