Radicals

Example 1: Finding the Domain of Square-root Functions Find the domain of the square-root function. The expression under the radical sign must be greater than or equal to 0. x– 4 ≥ 0 + 4 + 4 x ≥ 4 Solve the inequality. Add 4 to both sides. The domain is the set of all real numbers greater than or equal to 4.

Example 2: Finding the Domain of Square-root Functions Find the domain of the square-root function. The expression under the radical sign must be greater than or equal to 0. x + 3 ≥ 0 –3 –3 x ≥ –3 Solve the inequality. Subtract 3 from both sides. The domain is the set of all real numbers greater than or equal to –3.

Example 3: Finding the Domain of Square-root Functions Find the domain of the square-root function. The expression under the radical sign must be greater than or equal to 0. 2x – 1 ≥ 0 +1 +1 2x ≥ 1 Solve the inequality. Add 1 to both sides. Divide both sides by 2. The domain is the set of all real numbers greater than or equal to .

An expression that contains a radical sign is a radical expression An expression that contains a radical sign is a radical expression. There are many different types of radical expressions, but in this course, you will only study radical expressions that contain square roots. Examples of radical expressions: The expression under a radical sign is the radicand. A radicand may contain numbers, variables, or both. It may contain one term or more than one term.

Example 4: Using the Product Property of Square Roots Simplify. All variables represent nonnegative numbers. Factor the radicand using perfect squares. Product Property of Square Roots. Simplify.

Example 5: Using the Product Property of Square Roots Simplify. All variables represent nonnegative numbers. Product Property of Square Roots. Product Property of Square Roots. Since x is nonnegative, .

Example 6: Using the Product Property of Square Roots Simplify. All variables represent nonnegative numbers. Factor the radicand using perfect squares. Product Property of Square Roots. Simplify.

Example 7: Using the Quotient Property of Square Roots Simplify. All variables represent nonnegative numbers. A. B. Simplify. Quotient Property of Square Roots. Quotient Property of Square Roots. Simplify. Simplify.

Example 8: Using the Quotient Property of Square Roots Simplify. All variables represent nonnegative numbers. Quotient Property. Product Property. Write as . Simplify.

Example 9: Using the Quotient Property of Square Roots Simplify. All variables represent nonnegative numbers. Quotient Property. Simplify.

Add/Subtract Like Radicals

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

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

Example 1&2: Adding and Subtracting Square-Root Expressions Add or subtract. A. The terms are like radicals. B. The terms are unlike radicals. Do not combine.

Example 3&4: Adding and Subtracting Square-Root Expressions Add or subtract. C. the terms are like radicals. D. Identify like radicals. Combine like radicals.

Example 5: Simplify Before Adding or Subtracting Simplify each expression. Factor the radicands using perfect squares. Product Property of Square Roots. Simplify. Combine like radicals.

Example 6: Simplify Before Adding or Subtracting Simplify each expression. Factor the radicands using perfect squares. Product Property of Square Roots. Simplify. Combine like radicals.

Example 7: Geometry Application Find the perimeter of the triangle. Give the answer as a radical expression in simplest form. Write an expression for perimeter. Factor 20 using a perfect square. Product Property of Square Roots. Simplify. Combine like radicals.

Multiply and Divide Radicals You can use the Product and Quotient Properties of square roots you have already learned to multiply and divide expressions containing square roots.

Example 1: Multiplying Square Roots Multiply. Write the product in simplest form. Product Property of Square Roots. Multiply the factors in the radicand. Factor 16 using a perfect-square factor. Product Property of Square Roots Simplify.

Example 2: Multiplying Square Roots Multiply. Write the product in simplest form. Expand the expression. Commutative Property of Multiplication. Product Property of Square Roots. Simplify the radicand. Simplify the Square Root. Multiply.

Example 3: Multiplying Square Roots Multiply. Write the product in simplest form. Factor 4 using a perfect-square factor. Product Property of Square Roots. Take the square root.. Simplify.

Example 4: Using the Distributive Property Multiply. Write the product in simplest form. Distribute Product Property of Square Roots. Simplify the radicands. Simplify.

First terms Outer terms Inner terms Last terms See Lesson 7-7. Remember!

= 20 + 3

Example 5: Multiplying Sums and Differences of Radicals Multiply. Write the product in simplest form. Use the FOIL method. Simplify by combining like terms. Simplify the radicand. Simplify.

Example 6: Rationalizing the Denominator Simplify the quotient. Multiply by a form of 1 to get a perfect-square radicand in the denominator. Product Property of Square Roots. Simplify the denominator.

Example 7 Simplify the quotient. Multiply by a form of 1 to get a perfect-square radicand in the denominator. Simplify the square root in denominator.

Solving Radical Equations

Example 1: Solving Simple Radical Equations Solve the equation. Check your answer. Square both sides. x = 25 Check Substitute 25 for x in the original equation. 5 5  Simplify.

Example 2: Solving Simple Radical Equations Solve the equation. Check your answer. Square both sides. 100 = 2x 50 = x Divide both sides by 2. Check 10 10 Substitute 50 for x in the original equation.  Simplify.

Example 3: Solving Simple Radical Equations Solve the equation. Check your answer. Add 4 to both sides. Square both sides. x = 81 Check 9 – 4 5  5 5

Example 4: Solving Simple Radical Equations Solve the equation. Check your answer. Square both sides. x = 46 Subtract 3 from both sides. Check  7 7

Example 5: Solving Simple Radical Equations Solve the equation. Check your answer. Subtract 6 from both sides. Square both sides. 5x + 1 = 16 5x = 15 Subtract 1 from both sides. x = 3 Divide both sides by 5.

Example 6: Solving Radical Equations by Multiplying or Dividing Solve the equation. Check your answer. Method 1 Divide both sides by 4. Square both sides. x = 64

Example 7: Solving Radical Equations by Multiplying or Dividing Solve the equation. Check your answer. Method 1 Multiply both sides by 2. Square both sides. 144 = x

Example 8 Solve the equation. Check your answer. Method 1 Multiply both sides by 5. Square both sides. Divide both sides by 4. x = 100

Example 9: Solving Radical Equations with Square Roots on Both Sides Solve the equation. Check your answer. Square both sides. 2x – 1 = x + 7 Add 1 to both sides and subtract x from both sides. x = 8 Check 