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Nth Roots and Rational Exponents

How do you change a power to rational form and vice versa?

How do you evaluate radicals and powers with rational exponents?

How do you solve equations involving radicals and powers with rational exponents?

The Nth root

Index Number

Radicand

Radical

The index number becomes the denominator of the exponent.

n > 1

Radicals

If n is odd – one real root.

If n is even and

a > 0 Two real roots

a = 0 One real root

a < 0 No real roots

Example: Radical form to Exponential Form

Change to exponential form.

or

or

Example: Exponential to Radical Form

Change to radical form.

The denominator of the exponent becomes the index number of the radical.

Example: Evaluate Without a Calculator

Evaluate without a calculator.

Example: Solving an equation

Solve the equation:

Note: index number is even, therefore, two answers.

Rules

Rational exponents and radicals follow the properties of exponents.

Also, Product property for radicals

Quotient property for radicals

Example: Using the Quotient Property

Simplify.

Adding and Subtracting Radicals

Two radicals are like radicals, if they have the same index number and radicand

Example

Addition and subtraction is done with like radicals.

Example: Addition with like radicals

Simplify.

Note: same index number and same radicand.

Add the coefficients.

Example: Subtraction

Simplify.

Note: The radicands are not the same. Check to see if we can change one or both to the same radicand.

Note: The radicands are the same. Subtract coefficients.