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Rational Exponents Intermediate Algebra MTH04 When the

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Properties of Exponents
am • an = am+n
Product of Powers Property
Power of Power Property
(am)n = am•n
Power of Product Property
(ab)m = ambm
Negative Power Property
a-n = 1/an, aп‚№ 0
a0 = 1
Zero Power Property
Quotients of Powers Property
Power of Quotient Property
am
a
(
n
a
b
)
пЂЅ a
m
пЂЅ
mпЂ­ n
am
b
m
, a п‚№ 0
,b п‚№ 0
Intermediate Algebra -MTH04
Rational Exponents
Mr. Joyner
Tidewater Community College
Intermediate Algebra MTH04
Rational Exponents
Radicals (also called roots) are
directly related to exponents.
Intermediate Algebra MTH04
Rational Exponents
All radicals (roots) can be
written in a different format
without a radical symbol.
7.1 – Radicals
Radical Expressions
Finding a root of a number is the inverse operation of raising
a number to a power.
This symbol is the radical or the radical sign
radical sign
index
n
a
radicand
The expression under the radical sign is the radicand.
The index defines the root to be taken.
Intermediate Algebra MTH04
Rational Exponents
This different format uses a
rational (fractional) exponent.
Intermediate Algebra MTH04
Rational Exponents
When the exponent of the radicand
(expression under the radical symbol)
is one, the rational exponent form
of a
1
radical looks like this: n a пЂЅ a n
Remember that the index, n, is a whole number
equal to or greater than 2.
Intermediate Algebra MTH04
Rational Exponents
Examples:
1
6 пЂЅ 62
1
3
11 пЂЅ 11 3
base
• When a base has a fractional exponent, do not
think of the exponent in the same way as when it is
a whole number.
• When a base has a fractional exponent, the
exponent is telling you that you have a radical
written in a different form.
Intermediate Algebra MTH04
Rational Exponents
For any exponent of the radicand, the
rational exponent form of a radical
looks like this:
n
a
m
пЂЅ
пЂЁ aпЂ©
n
m
m
пЂЅ a
n
Intermediate Algebra MTH04
Rational Exponents
1
How do you simplify 16 2
?
• Reduce the rational exponent, if possible.
• You can rewrite the expression using a radical.
• Simplify the radical expression, if possible.
• Write your answer in simplest form.
Intermediate Algebra MTH04
Rational Exponents
1
16
2
пЂЅ
16 пЂЅ 4
Example:
1
125
3
пЂЅ
3
125 пЂЅ 5
Intermediate Algebra MTH04
Rational Exponents
Examples:
пЂЅ
пЂЁ
пЂЅ
пЂЁ
2
32
5
5
64
3
5
3
32
пЂ©
64
пЂ©
2
5
пЂЅ пЂЁ2 пЂ© пЂЅ 4
2
пЂЅ
пЂЁ4 пЂ©
5
пЂЅ 1024
Intermediate Algebra MTH04
Rational Exponents
Examples:
1
пЂЁ пЂ­ 16 пЂ© 2
2
пЂЁ пЂ­ 216 пЂ© 3
пЂ­ 16 пЂЅ
пЂЅ
пЂЅ
пЂЁ
3
пЂ­ 216
No real number solution
пЂ©
2
пЂЅ пЂЁ пЂ­ 6 пЂ© пЂЅ 36
2
Rational Exponents
More Examples:
пѓ¦ 1 пѓ¶
пѓ§
пѓ·
пѓЁ 27 пѓё
2
2
3
3
3
1
2
3
3
27
1
2
27
3
3
2
1
1
729
9
or
пѓ¦ 1 пѓ¶
пѓ§
пѓ·
пѓЁ 27 пѓё
2
2
3
1
3
2
27
пЂЁ1пЂ©
3
3
пЂЁ
3
27
2
пЂ©
пЂЁ1 пЂ©
2
пЂЁ3 пЂ©
2
2
1
9
Intermediate Algebra MTH04
Rational Exponents
The basic properties for integer
exponents also hold for rational
exponents as long as the expression
represents a real number.
See the chart on page 389 of your
text.
Intermediate Algebra MTH04
Rational Exponents
Example:
2
5
3
1
пЂЅ 5
пѓ¦2 1пѓ¶
пѓ§ пЂ­ пѓ·
пѓЁ3 2пѓё
пЂЅ 5
пѓ¦4 3пѓ¶
пѓ§ пЂ­ пѓ·
пѓЁ6 6пѓё
1
пЂЅ 56
52
What would the answer above be if you
were to write it in radical form?
Intermediate Algebra MTH04
Rational Exponents
Example:
2
5
3
1
52
пЂЅ 5
пѓ¦2 1пѓ¶
пѓ§ пЂ­ пѓ·
пѓЁ3 2пѓё
пЂЅ 5
пѓ¦4 3пѓ¶
пѓ§ пЂ­ пѓ·
пѓЁ6 6пѓё
1
пЂЅ 5
6
пЂЅ
6
5
Intermediate Algebra MTH04
Rational Exponents
Do you remember the basic Rules of
Exponents that you learned in Roots
and Radicals?
See the next two slides for a quick
review.
Intermediate Algebra MTH04
Rational Exponents
The Square Root Rules (Properties)
Multiplication
aпѓ— b пЂЅ
a пѓ—b
Division
a
b
пЂЅ
a
b
b may not be equal to 0.
Intermediate Algebra MTH04
Rational Exponents
The Cube Root Rules (Properties)
Multiplication
3
aпѓ— b пЂЅ
3
3
a пѓ—b
Division
3
3
a
b
пЂЅ
3
a
b
b may not be equal to 0.
Intermediate Algebra MTH04
Rational Exponents
The more general rules for any
radical are as follows …
Intermediate Algebra MTH04
Rational Exponents
The Rules (Properties)
Multiplication
n
aпѓ— b пЂЅ
n
n
a пѓ—b
Division
n
n
a
b
пЂЅ
n
a
b
b may not be equal to 0.
Intermediate Algebra MTH04
Rational Exponents
These same rules in rational exponent
form are as follows …
Intermediate Algebra MTH04
Rational Exponents
The Rules (Properties)
Multiplication
1
1
1
a n пѓ— b n пЂЅ пЂЁa пѓ— b пЂ© n
Division
1
1
an
пѓ¦ a пѓ¶n
пЂЅ пѓ§ пѓ·
пѓЁbпѓё
1
bn
b may not be equal to 0.
Intermediate Algebra MTH04
Rational Exponents
In working with radicals, whether in radical
form or in fractional exponent form, simplify
wherever and whenever possible.
What is the process for simplifying radical
expressions?
Intermediate Algebra MTH04
Rational Exponents
Simplifying radicals – A radical expression is in
simplest form once ALL of the following
conditions have been met.…
• the radicand (expression under the radical symbol) cannot
be written in an exponent form with any factor having an
exponent equal to or larger than the index of the radical;
• there is no fraction under the radical symbol;
• there is no radical in a denominator.
Intermediate Algebra MTH04
Rational Exponents
Examples – Simplifying Radical Expressions:
3
54 пЂЅ
3
3
пЂЅ
5
8
3
6
пЂЅ
27 пѓ— 2 пЂЅ
3
5
пѓ—
5
8
3
6
15
пЂЅ
5
3
пѓ—
3
6
2
6
2
27 пѓ—
3
3
2 пЂЅ 3 2
3
15
пЂЅ
5
25
пЂЅ
8
3
3
6
6
3
2
пЂЅ
8
3
6
6
2
пЂЅ
4
3
6
3
2
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