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WRITING RADICALS IN RATIONAL WRITING 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
ma BASEBASE
EXPONENTEXPONENT
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THE EXPONENTTHE EXPONENT
ma 322 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 na 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 na a
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EXAMPLE 1EXAMPLE 1
Evaluate 43/2 in radical form and simplify.
/m
m n na a
33/24 4
3 34 2
8
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EXAMPLE 2EXAMPLE 2Evaluate 41/2 in radical form and
simplify.
2
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YOUR TURNYOUR TURNEvaluate (–27)2/3 in radical form and
simplify.
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EXAMPLE 3EXAMPLE 3
Evaluate –274/3 in radical form and simplify.
/m
m n na a
43 27
Hint: Remember, the negative is OUTSIDE of the base
81
Use calculator to check
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EXAMPLE 4EXAMPLE 4Evaluate in radical form and simplify.
35 4 1x
3/54 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
//
1mm n n
m na a
a
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EXAMPLE 5EXAMPLE 5
Evaluate (27)–2/3 in radical form and simplify.
1
9
2/327
23 27
23
1
27
21
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
1/225
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 na a a ( ) m n m na a
( ) m m mab a b( )
1nn
aa
mm n
n
aa
a
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EXAMPLEEXAMPLE 7 7Simplify
• Saying goes: BASE, BASE, ADD
If the BASES are the same, ADD the powersm n m na a a
4 52 2
92
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EXAMPLE 8EXAMPLE 8Simplify 1/2 1/3x x
5/6x
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YOUR TURNYOUR TURNSimplify 3/5 1/4x x
17/20x
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EXAMPLE 9EXAMPLE 9Simplify
• Saying goes: POWER, POWER, MULTIPLY
If the POWERS are near each other, MULTIPLY the powers – usually deals with PARENTHESES
( ) m n m na a
542
202
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EXAMPLE 10EXAMPLE 10Simplify
( ) m m mab a b
52/52x
232x
/ ( / )( )52 5 5 2 5 52 2x x
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YOUR TURNYOUR TURNSimplify 41/4 2/33x y
2/3
81x
y
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EXAMPLE 11EXAMPLE 11Simplify
• Saying goes: When dividing an expression with a power, SUBTRACT the powers.
m
m nn
aa
a
1/3
4/3
7
7
1
7
// /
/
1 31 3 4 3
4 3
77
7
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EXAMPLE 12EXAMPLE 12Simplify
3
x
x
1/6x
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EXAMPLE 13EXAMPLE 13Simplify
33 4
5
xx x
x
13/60x