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Rational Exponents Mr. Joyner Tidewater Community College Intermediate Algebra
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Rational ExponentsMr. Joyner
Tidewater Community College
Intermediate Algebra
Intermediate Algebra
Radicals (also called roots) are directly related to exponents.
Rational Exponents
Intermediate Algebra
All radicals (roots) can be written in a different format without a radical symbol.
Rational Exponents
Intermediate Algebra
This different format uses a rational (fractional) exponent.
Rational Exponents
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
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
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
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
Intermediate Algebra
Rational Exponents
Example:
2
1
16 416
3
1
125 51253
Intermediate Algebra
Rational Exponents
Examples:
5
2
32 4232 225
3
5
64 1024464 553
Intermediate Algebra
Rational Exponents
Examples:
No real number solution 2
1
16 16
3
2
216 366216 223
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.
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
Intermediate Algebra
Rational Exponents
Example:
6
16
3
6
4
2
1
3
2
2
1
3
2
555
5
5
6 5
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.
Multiplication Division
b may not be equal to 0.
Intermediate Algebra
The Square Root Rules (Properties)
Rational Exponents
b
a
b
ababa
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
Intermediate Algebra
Rational Exponents
The more general rules for any radical are as follows …
Multiplication Division
b may not be equal to 0.
Intermediate Algebra
The Rules (Properties)
Rational Exponents
nn
n
b
a
b
annn baba
Intermediate Algebra
Rational Exponents
These same rules in rational exponent form are as follows …
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
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?
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.
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
