1. Properties of Exponents
Product of Powers Property
am • an = am+n
Power of Power Property
(am)n = am•n
Power of Product Property
(ab)m = ambm
Negative Power Property
a-n = 1/an, a 0
Zero Power Property
a0 = 1
Quotients of Powers Property
Power of Quotient Property

2. Intermediate Algebra -MTH04 Tidewater Community College
Rational Exponents
Mr. Joyner
Tidewater Community College

3. Intermediate Algebra MTH04
Rational Exponents
Radicals (also called roots) are directly related to exponents.

4. Intermediate Algebra MTH04
Rational Exponents
All radicals (roots) can be written in a different format without a radical symbol.

5. 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
radicand
The expression under the radical sign is the radicand.
The index defines the root to be taken.

6. Intermediate Algebra MTH04
Rational Exponents
This different format uses a rational (fractional) exponent.

7. Intermediate Algebra MTH04
Rational Exponents
When the exponent of the radicand (expression under the radical symbol) is one, the rational exponent form of a radical looks like this:
Remember that the index, n, is a whole number equal to or greater than 2.

8. Intermediate Algebra MTH04
Rational Exponents
Examples:
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.

9. Intermediate Algebra MTH04
Rational Exponents
For any exponent of the radicand, the rational exponent form of a radical looks like this:

10. Intermediate Algebra MTH04
Rational Exponents
How do you simplify ?
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.

11. Intermediate Algebra MTH04
Rational Exponents
Example:

12. Intermediate Algebra MTH04
Rational Exponents
Examples:

13. Intermediate Algebra MTH04
Rational Exponents
Examples:
No real number solution

14. Rational Exponents
More Examples: or

15. 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.

16. Intermediate Algebra MTH04
Rational Exponents
Example:
What would the answer above be if you were to write it in radical form?

17. Intermediate Algebra MTH04
Rational Exponents
Example:

18. 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.

19. Intermediate Algebra MTH04
Rational Exponents
The Square Root Rules (Properties)
Multiplication
Division
b may not be equal to 0.

20. Intermediate Algebra MTH04
Rational Exponents
The Cube Root Rules (Properties)
Multiplication
Division
b may not be equal to 0.

21. Intermediate Algebra MTH04
Rational Exponents
The more general rules for any radical are as follows …

22. Intermediate Algebra MTH04
Rational Exponents
The Rules (Properties)
Multiplication
Division
b may not be equal to 0.

23. Intermediate Algebra MTH04
Rational Exponents
These same rules in rational exponent form are as follows …

24. Intermediate Algebra MTH04
Rational Exponents
The Rules (Properties)
Multiplication
Division
b may not be equal to 0.

25. 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?

26. 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.

27. Intermediate Algebra MTH04
Rational Exponents
Examples – Simplifying Radical Expressions: