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 0 45 5 9 5 _ ab 2 5 a b 2 5 3 5 5 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 37 64 5 _ 37 64 x 5 x y 2 y 2 5 _ 37 8 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. 3 5 3 3 3 3 c d 5 c d 3 d d 5 3 3 5 _ c d d The distributive property can be used to combine like terms of radical expressions. 4 0 3 2 9 0 5 4 0 2 9 0 3 5 (4 2 9) 0 3 5 25 0 3 5 (4 2 20 ) 5 4 5 2 5 20 5 4 5 2 _ (5 20) 5 4 5 2 _ 00 5 4 5 2 0 36 Unit : Relationships between Quantities and Expressions
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. 24 5 4 6 2 Use the product property of radicals. 24 5 4 6 5 4 6 5 2 6 24 5 2 6 Simplify expression B. Simplify the fraction under the radical. The GCF of the numerator and the denominator is 2. DISCUSS 50 5 9 25 So, 50 5 9 25. 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. 9 25 5 _ 9 25 5 3 5 50 5 3 5 Lesson 6: Radical Expressions 37
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. 6 5 38 Unit : Relationships between Quantities and Expressions
EXAMPLE B Simplify the expression. ( 7 3 )( 7 2 2 3 ) Use the distributive property. Multiply the first term in the left factor by each term in the right factor. 7 ( 7 2 2 3 ) 5 7 7 2 7 2 3 5 7 2 2 _ 7 3 5 7 2 2 2 Multiply the second term in the left factor by each term in the right factor. 3 ( 7 2 2 3 ) 5 3 7 2 3 2 3 5 _ 3 7 2 2 3 5 2 2 6 2 Simplify the product. CHECK Write the product. ( 7 3 ) ( 7 2 2 3 ) 5 7 2 2 2 2 2 6 Combine like terms. 7 2 2 2 2 2 6 5 7 2 6 2 2 2 2 5 2 2 Simplified, ( 7 3 ) ( 7 2 2 3 ) 5 2 2. Reverse the order of the factors in each expression and use the distributive property to find the product. Lesson 6: Radical Expressions 39
Practice Simplify each expression. REMEMBER Look for perfect squares to apply the product and quotient properties to square roots.. 40 2. _ 25 3. _ 72a 4. 3 4 5. 2 27 _ 6. _ 36 400 Rationalize the denominator in each expression. 7. 2 3 8. 64 x 9. y 5 HINT Rationalize the denominator by multiplying the numerator and denominator by the radical in the denominator. Simplify each expression. 0. 8 23 5 23. 2 7 2 9 2 6 7 2. 4 2 2 3 3. 2 35 2 0 6 35 _ 00 4. 9 0 4 3 2 3 0 2 9 3 40 Unit : Relationships between Quantities and Expressions
Simplify each product. 5. 2 ( 8 5 ) 6. 3 ( 6 ) 7. 0 ( 5 2 8 ) 8. 3 ( 9 2 2 ) 9. 5 ( 2 5 2 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 )( 2 3 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