7 17 2 Nth Roots and Rational Exponents

7. 1/7. 2 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 Radical Index Number n>1 Radicand The index number becomes the denominator of the exponent.

Radicals • If n is odd – one real root. • If n is even and – a>0 – a=0 – a<0 Two real roots One real root 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.