rational exponents mr. joyner tidewater community college intermediate algebra
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Rational ExponentsMr. Joyner
Tidewater Community College
Intermediate Algebra
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Intermediate Algebra
Radicals (also called roots) are directly related to exponents.
Rational Exponents
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Intermediate Algebra
All radicals (roots) can be written in a different format without a radical symbol.
Rational Exponents
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Intermediate Algebra
This different format uses a rational (fractional) exponent.
Rational Exponents
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Intermediate Algebra
When the exponent of the radicand (expression under the radical symbol) is one, the rational exponent form of a radical looks like this:
Rational Exponents
Remember that the index, n, is a whole number equal to or greater than 2.
nn aa1
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Intermediate Algebra
Rational Exponents
Examples:
• 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.
2
1
66 3
13 1111
base
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Intermediate Algebra
Rational Exponents
For any exponent of the radicand, the rational exponent form of a radical looks like this:
n
mm
nn m aaa
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How do you simplify ?
Intermediate Algebra
Rational Exponents
• You can rewrite the expression using a radical.
• Simplify the radical expression, if possible.
• Write your answer in simplest form.
• Reduce the rational exponent, if possible.
2
1
16
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Intermediate Algebra
Rational Exponents
Example:
2
1
16 416
3
1
125 51253
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Intermediate Algebra
Rational Exponents
Examples:
5
2
32 4232 225
3
5
64 1024464 553
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Intermediate Algebra
Rational Exponents
Examples:
No real number solution 2
1
16 16
3
2
216 366216 223
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Intermediate Algebra
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.
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Intermediate Algebra
Rational Exponents
Example:
What would the answer above be if you were to write it in radical form?
6
16
3
6
4
2
1
3
2
2
1
3
2
555
5
5
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Intermediate Algebra
Rational Exponents
Example:
6
16
3
6
4
2
1
3
2
2
1
3
2
555
5
5
6 5
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Intermediate Algebra
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.
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Multiplication Division
b may not be equal to 0.
Intermediate Algebra
The Square Root Rules (Properties)
Rational Exponents
b
a
b
ababa
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Multiplication Division
b may not be equal to 0.
Intermediate Algebra
The Cube Root Rules (Properties)
Rational Exponents
33
3
b
a
b
a333 baba
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Intermediate Algebra
Rational Exponents
The more general rules for any radical are as follows …
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Multiplication Division
b may not be equal to 0.
Intermediate Algebra
The Rules (Properties)
Rational Exponents
nn
n
b
a
b
annn baba
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Intermediate Algebra
Rational Exponents
These same rules in rational exponent form are as follows …
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Multiplication Division
b may not be equal to 0.
Intermediate Algebra
The Rules (Properties)
Rational Exponents
nnn baba111
n
n
n
b
a
b
a1
1
1
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Intermediate Algebra
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?
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Intermediate Algebra
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.
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Intermediate Algebra
Rational Exponents
Examples – Simplifying Radical Expressions:
3 54 3 227 33 227 3 23
5
3
3 6
8
5
5
5
3
25
155
15
3 2
3 2
3 6
6
6
8
3 3
3 2
6
68 6
68 3 2
3
64 3 2