Multiplying and Dividing Expressions with Radicals

New York State Common Core Math Geometry, Module 2, Lesson 22

Worksheets for Geometry

Student Outcomes

• Students multiply and divide expressions that contain radicals to simplify their answers.
• Students rationalize the denominator of a number expressed as a fraction.

Multiplying and Dividing Expressions with Radicals

Classwork

Exercises 1–5

Simplify as much as possible.

1. √17² =
   
2. √5¹⁰ =
   
3. √4𝑥⁴ =
   
4. Complete parts (a) through (c).
   a. Compare the value of √36 to the value of √9 × √4
   b. Make a conjecture about the validity of the following statement: For nonnegative real numbers 𝑎 and 𝑏, √𝑎𝑏 = √𝑎 ∙ √𝑏. Explain.
   c. Does your conjecture hold true for 𝑎 = −4 and 𝑏 = −9?
   
5. Complete parts (a) through (c).
   a. Compare the value of √100/25 to the value of √100/√25.
   b. Make a conjecture about the validity of the following statement: For nonnegative real numbers 𝑎 and 𝑏, when 𝑏 ≠ 0, √𝑎𝑏 = √𝑎√𝑏. Explain.
   c. Does your conjecture hold true for 𝑎 = −100 and 𝑏 = −25?

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Exercises 6–17

Simplify each expression as much as possible, and rationalize denominators when applicable.

17. The captain of a ship recorded the ship’s coordinates, then sailed north and then west, and then recorded the new coordinates. The coordinates were used to calculate the distance they traveled, √578 km. When the captain asked how far they traveled, the navigator said, “About 24 km.” Is the navigator correct? Under what conditions might he need to be more precise in his answer?

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