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Rational Exponents

Radicals (also called roots) are directly related to exponents.

Rational Exponents

All radicals (roots) can be written in a different format without a radical symbol.

Rational Exponents

This different format uses a rational (fractional) exponent.

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.

Intermediate Algebra MTH04

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.

Rational Exponents

For any exponent of the radicand, the rational exponent form of a radical looks like this:

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.

Rational Exponents

The basic properties for integer exponents also hold for rational exponents as long as the expression represents a real number.

Rational Exponents

Example:

What would the answer above be if you were to write it in radical form?

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.

Rational Exponents

The Square Root Rules (Properties)

Multiplication

Division

b may not be equal to 0.

Rational Exponents

The Cube Root Rules (Properties)

Multiplication

Division

b may not be equal to 0.

Rational Exponents

The more general rules for any radical are as follows …

Rational Exponents

The Rules (Properties)

Multiplication

Division

b may not be equal to 0.

Rational Exponents

These same rules in rational exponent form are as follows …

Rational Exponents

The Rules (Properties)

Multiplication

Division

b may not be equal to 0.

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?

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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