Explanation of topic exponents and radicals

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WRITING RADICALS IN RATIONALWRITING RADICALS IN RATIONAL
FORMFORM
Section 10.2

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DEFINITIONSDEFINITIONS
Base: The term/variable of which is being raised
upon
Exponent: The term/variable is raised by a term. AKA
Power
m
a BASEBASE
EXPONENTEXPONENT

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THE EXPONENTTHE EXPONENT
m
a 3
2
2 2 2= × ×
2 3≠ ×
2 2 2= × ×

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NTH ROOT RULENTH ROOT RULE
• M is the power (exponent)
• N is the root
• A is the base
( )=/
m
m n n
a a

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RULESRULES
Another way of writing is 251/2
.
is written in radical expression form.
251/2
is written in rational exponent form.
Why is square root of 25 equals out of 25 raised to
½ power?
25
25

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EXAMPLE 1EXAMPLE 1
Evaluate 43/2
in radical form and simplify.
( )=/
m
m n n
a a

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EXAMPLE 1EXAMPLE 1
Evaluate 43/2
in radical form and simplify.
( )=/
m
m n n
a a
( )
3
3/2
4 4=
( )
3
3
4 2=
8

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EXAMPLE 2EXAMPLE 2
Evaluate 41/2
in radical form and
simplify.
2

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YOUR TURNYOUR TURN
Evaluate (–27)2/3
in radical form and
simplify.

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10
EXAMPLE 3EXAMPLE 3
Evaluate –274/3
in radical form and
simplify.
( )=/
m
m n n
a a
( )
4
3
27−
Hint: Remember, the negative is OUTSIDE of the base
81−
Use calculator
to check

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11
EXAMPLE 4EXAMPLE 4
Evaluate in radical form and simplify.
( )
35
4 1x −
( )
3/5
4 1x −

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NTH ROOT RULENTH ROOT RULE
• M is the power (exponent)
• N is the root
• A is the base
DROP AND SWAP
( )
−
−
= =/
/
1m
m n n
m n
a a
a

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EXAMPLE 5EXAMPLE 5
Evaluate (27)–2/3
in radical form and simplify.
1
9
( )
2/3
27
−
= ( )
2
3
27
−
=
( )
2
3
1
27
=
( )
2
1
3

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EXAMPLE 6EXAMPLE 6
Evaluate (–64)–2/3
in radical form and simplify.
1
16

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YOUR TURNYOUR TURN
Evaluate in radical form and simplify.
( )
( )
( )
( )
1 1
1 1
25 36
36 25
−
−
 
 ÷= =
 ÷
 ÷
 
1/2
25
36
−
 
 ÷
 
6
5

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PROPERTIES OF EXPONENTSPROPERTIES OF EXPONENTS
Product of a Power:
Power of a Power:
Power of a Product:
Negative Power Property:
Quotient Power Property:
m n m n
a a a +
× =
( ) ×
=m n m n
a a
( ) =m m m
ab a b
( )−
=
1n
n
a
a
−
=
m
m n
n
a
a
a

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EXAMPLEEXAMPLE 77
Simplify
• Saying goes: BASE, BASE, ADD
If the BASES are the same, ADD the powers
m n m n
a a a +
× =
4 5
2 2×
9
2

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EXAMPLE 8EXAMPLE 8
Simplify 1/2 1/3
x x×
5/6
x

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YOUR TURNYOUR TURN
Simplify 3/5 1/4
x x×
17/20
x

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EXAMPLE 9EXAMPLE 9
Simplify
• Saying goes: POWER, POWER, MULTIPLY
If the POWERS are near each other, MULTIPLY
the powers – usually deals with
PARENTHESES
( ) ×
=m n m n
a a
( )
5
4
2
20
2

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EXAMPLE 10EXAMPLE 10
Simplify
( ) =m m m
ab a b
( )
5
2/5
2x
2
32x
( )/ ( / )( )
=
52 5 5 2 5 5
2 2x x

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YOUR TURNYOUR TURN
Simplify ( )
4
1/4 2/3
3x y−
2/3
81x
y

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EXAMPLE 11EXAMPLE 11
Simplify
• Saying goes: When dividing an expression
with a power, SUBTRACT the powers.
−
=
m
m n
n
a
a
a
1/3
4/3
7
7
1
7
/
/ /
/
−
=
1 3
1 3 4 3
4 3
7
7
7

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EXAMPLE 12EXAMPLE 12
Simplify
3
x
x
1/6
x

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EXAMPLE 13EXAMPLE 13
Simplify ( )
3
3 4
5
x
x x
x
×
13/60
x