Radical Expressions LESSON. 36 Unit 1: Relationships between Quantities and Expressions

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1 LESSON 6 Radical Expressions UNDERSTAND You can use the following to simplify radical expressions. Product property of radicals: The square root of a product is equal to the square root of the factors. ab 5 a b where a 0 and b _ ab 2 5 a b b a Quotient property of radicals: The square root of a quotient is equal to the quotient of the square roots of the numerator and the denominator. a b 5 a where a 0 and b 0 b _ x 5 x y 2 y 2 5 _ x y Rationalize the denominator: To be in simplest form, the denominator of a fraction should not be irrational. Therefore, the denominator of a fraction should not contain a radical. To rationalize the denominator, convert the fraction into a form where the denominator has only rational (fractional or whole number) values c d 5 c d 3 d d _ c d d The distributive property can be used to combine like terms of radical expressions (4 2 9) ( ) _ (5 20) _ Unit : Relationships between Quantities and Expressions

2 Connect Simplify the expressions. A. 24 B. 50 Simplify expression A. Using the product property of radicals, write 24 as a product in which one of the factors is a perfect square Use the product property of radicals Simplify expression B. Simplify the fraction under the radical. The GCF of the numerator and the denominator is 2. DISCUSS So, If you start by using the quotient property of radicals to simplify the expression 8 50, will you get the same answer? Explain. 2 Use the quotient property of radicals _ Lesson 6: Radical Expressions 37

3 EXAMPLE A Simplify the expression. 3 5b Use the quotient property of radicals. 3 5b 5 _ 3 5b 2 Rationalize the denominator. Multiply the numerator and the denominator by 5b. _ 5b 5 _ 3 5b _ 5b 5b 3 5 5b 25b 2 _ 3 Use the product property of radicals. Write 25b 2 as a product in which one of the factors is a perfect square. _ 5b _ 5 5b 25b 2 25 b 2 5 _ 5b 5b _ Simplified, 3 5b 5 _ 5b 5b. TRY Simplify the expression Unit : Relationships between Quantities and Expressions

4 EXAMPLE B Simplify the expression. ( 7 3 )( ) Use the distributive property. Multiply the first term in the left factor by each term in the right factor. 7 ( ) _ Multiply the second term in the left factor by each term in the right factor. 3 ( ) _ Simplify the product. CHECK Write the product. ( 7 3 ) ( ) Combine like terms Simplified, ( 7 3 ) ( ) Reverse the order of the factors in each expression and use the distributive property to find the product. Lesson 6: Radical Expressions 39

5 Practice Simplify each expression. REMEMBER Look for perfect squares to apply the product and quotient properties to square roots _ _ 72a _ 6. _ Rationalize the denominator in each expression x 9. y 5 HINT Rationalize the denominator by multiplying the numerator and denominator by the radical in the denominator. Simplify each expression _ Unit : Relationships between Quantities and Expressions

6 Simplify each product ( 8 5 ) 6. 3 ( 6 ) 7. 0 ( ) 8. 3 ( ) 9. 5 ( ) Solve. 20. EXPLAIN What properties must you use to simplify the expression 7 2? 2. APPLY A rectangular field is _ 28 feet long. It is _ 08 feet wide. What is the area of the field? Explain your thinking. 22. COMPARE Use the distributive property to find the product of ( 2 5 )( )and the product of ( a b )( a 2 b ). How are the products similar and how are they different? Lesson 6: Radical Expressions 4