Operations With Radical Expressions Section 10-3

Goals Goal Rubric To simplify sums and differences of radical expressions. To simplify products and quotients of radical expressions. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Like Radicals Unlike Radicals Conjugates

Like Radicals Square-root expressions with the same radicand are examples of like radicals.

Like Radicals Like radicals can be combined by adding or subtracting. You can use the Distributive Property to show how this is done: Notice that you can combine like radicals by adding or subtracting the numbers multiplied by the radical and keeping the radical the same.

Combining like radicals is similar to combining like terms. Helpful Hint

Example: Add or subtract. A. The terms are like radicals. B. The terms are unlike radicals. Do not combine.

Example: Add or subtract. the terms are like radicals. C. D. Identify like radicals. Combine like radicals.

Your Turn: Add or subtract. a. b. The terms are like radicals.

Your Turn: Add or subtract. c. d. The terms are like radicals. Combine like radicals.

More on Like Radicals Sometimes radicals do not appear to be like until they are simplified. Simplify all radicals in an expression before trying to identify like radicals and combining like radicals.

Example: Simplify each expression. Factor the radicands using perfect squares. Product Property of Square Roots. Simplify. Combine like radicals.

Example: Simplify each expression. Factor the radicands using perfect squares. Product Property of Square Roots. Simplify. The terms are unlike radicals. Do not combine.

Example: Simplify each expression. Factor the radicands using perfect squares. Product Property of Square Roots. Simplify. Combine like radicals.

Your Turn: Simplify each expression. Factor the radicands using perfect squares. Product Property of Square Roots. Simplify. Combine like radicals.

Your Turn: Simplify each expression. Factor the radicands using perfect squares. Product Property of Square Roots. Simplify. The terms are unlike radicals. Do not combine.

Your Turn: Simplify each expression. Factor the radicands using perfect squares. Product Property of Square Roots. Simplify. Combine like radicals.

Rationalizing Denominators Containing Two Terms If the denominator has two terms with one or more square roots, we can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator. A + B and A – B are conjugates. The product (A + B)(A – B) = A2 – B2

Example: Rationalize the denominator: The conjugate of the denominator is If we multiply the numerator and the denominator by the simplified denominator will not contain a radical. Therefore, we multiply by 1, choosing Multiply by 1.

Example: Continued Multiply by 1. Evaluate the exponents. Subtract. 3 Divide the numerator and denominator by 4. 1

Example: Continued On Rationalizing the Denominator… Simplify. If the denominator is a single square root term with nth root: Multiply numerator and denominator by that square root expression. If the denominator contains two terms involving square roots: Rationalize the denominator by multiplying the numerator and the denominator bythe conjugate of the denominator.

Rationalize the denominator: Example: Rationalize the denominator: Multiply numerator and denominator by the conjugate. (A+B)(A-B) = A2 – B2 Simplify.

Your Turn: Rationalize the denominator:

Your Turn: Simplify each expression. 1. 2.

Assignment 10-3 Exercises Pg. 621 - 622: #10 – 44 even