11-6 Radical Expressions Warm Up Lesson Presentation Lesson Quiz Holt Algebra 1

Warm Up Identify the perfect square in each set. 1. 45 81 27 111 2. 156 99 8 25 3. 256 84 12 1000 4. 35 216 196 72 81 25 256 196

Warm Up Continued Write each number as a product of prime numbers. 5. 36 6. 64 7. 196 8. 24

Objective Simplify radical expressions.

Vocabulary radical expression radicand

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.

Remember that positive numbers have two square roots, one positive and one negative. However, indicates a nonnegative square root. When you simplify, be sure that your answer is not negative. To simplify you should write because you do not know whether x is positive or negative.

Example 1: Simplifying Square-Root Expressions Simplify each expression. A. B. C.

Check It Out! Example 1 Simplify each expression. a. b.

Check It Out! Example 1 Simplify each expression. d. c.

Example 2A: 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 2B: 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, .

Check It Out! Example 2a Simplify. All variables represent nonnegative numbers. Factor the radicand using perfect squares. Product Property of Square Roots. Simplify.

Check It Out! Example 2b Simplify. All variables represent nonnegative numbers. Product Property of Square Roots. Product Property of Square Roots. Since y is nonnegative, .

Check It Out! Example 2c Simplify. All variables represent nonnegative numbers. Factor the radicand using perfect squares. Product Property of Square Roots. Simplify.

Example 3: 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.

Check It Out! Example 3 Simplify. All variables represent nonnegative numbers. a. b. Quotient Property of Square Roots. Simplify. Quotient Property of Square Roots. Simplify. Simplify.

Check It Out! Example 3c Simplify. All variables represent nonnegative numbers. Quotient Property of Square Roots. Factor the radicand using perfect squares. Simplify.

Example 4A: Using the Product and Quotient Properties Together Simplify. All variables represent nonnegative numbers. Quotient Property. Product Property. Write 108 as 36(3). Simplify.

Example 4B: Using the Product and Quotient Properties Together Simplify. All variables represent nonnegative numbers. Quotient Property. Product Property. Simplify.

Check It Out! Example 4a Simplify. All variables represent nonnegative numbers. Quotient Property. Product Property. Write 20 as 4(5). Simplify.

Check It Out! Example 4b Simplify. All variables represent nonnegative numbers. Quotient Property. Product Property. Write as . Simplify.

Check It Out! Example 4c Simplify. All variables represent nonnegative numbers. Quotient Property. Simplify.

Example 5: Application A quadrangle on a college campus is a square with sides of 250 feet. If a student takes a shortcut by walking diagonally across the quadrangle, how far does he walk? Give the answer as a radical expression in simplest form. Then estimate the length to the nearest tenth of a foot. 250 Quadrangle The distance from one corner of the square to the opposite one is the hypotenuse of a right triangle. Use the Pythagorean Theorem: c2 = a2 + b2.

Example 5 Continued Solve for c. Substitute 250 for a and b. Simplify. Factor 125,000 using perfect squares.

Example 5 Continued Use the Product Property of Square Roots. Simplify. Use a calculator and round to the nearest tenth. The distance is ft, or about 353.6 feet.

Check It Out! Example 5 A softball diamond is a square with sides of 60 feet. How long is a throw from third base to first base in softball? Give the answer as a radical expression in simplest form. Then estimate the length to the nearest tenth of a foot. 60 60 The distance from one corner of the square to the opposite one is the hypotenuse of a right triangle. Use the Pythagorean Theorem: c2 = a2 + b2.

Check It Out! Example 5 Continued Solve for c. Substitute 60 for a and b. Simplify. Factor 7,200 using perfect squares.

Check It Out! Example 5 Continued Use the Product Property of Square Roots. Simplify. Use a calculator and round to the nearest tenth. The distance is , or about 84.9 feet.

Lesson Quiz: Part I Simplify each expression. 1. 6 2. |x + 5| Simplify. All variables represent nonnegative numbers. 3. 4. 5. 6.

Lesson Quiz: Part II 7. Two archaeologists leave from the same campsite. One travels 10 miles due north and the other travels 6 miles due west. How far apart are the archaeologists? Give the answer as a radical expression in simplest form. Then estimate the distance to the nearest tenth of a mile. mi; 11.7mi