7 1 7 2 nth roots and rational exponents
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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.

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7 1 7 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
The Nth root

Radical

Index Number

n > 1

The index number becomes the denominator of the exponent.

Radicand

radicals
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
Example: Radical form to Exponential Form

Change to exponential form.

or

or

example exponential to radical form
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
Example: Evaluate Without a Calculator

Evaluate without a calculator.

example solving an equation
Example: Solving an equation

Solve the equation:

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

rules
Rules
  • Rational exponents and radicals follow the properties of exponents.
  • Also, Product property for radicals
  • Quotient property for radicals
adding and subtracting radicals
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
Example: Addition with like radicals

Simplify.

Note: same index number and same radicand.

Add the coefficients.

example subtraction
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.

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