1Add and subtract 2Multiply radical

Transcription

1 Then You simplified radical expressions. (Lesson 10-2) Now 1Add and subtract radical expressions. 2Multiply radical expressions. Operations with Radical Expressions Why? Conchita is going to run in her neighborhood to get ready for the soccer season. She plans to run the course that she has laid out three times each day. How far does Conchita have to run to complete the course that she laid out? How far does she run every day? x 2x x Virginia i SOL A. The student will express the square roots and cube roots of whole numbers and the square root of a monomial algebraic expression in simplest radical form. Add or Subtract Radical Expressions To add or subtract radical 1 expressions, the radicands must be alike in the same way that monomial terms must be alike to add or subtract. Monomials Radical Expressions a + 2a = ( + 2)a + 2 = ( + 2) = 6a = 6 9b - 2b = (9-2)b 9-2 = (9-2) = 7b = 7 Notice that when adding and subtracting radical expressions, the radicand does not change. This is the same as when adding or subtracting monomials. Example 1 Add and Subtract Expressions with Like Radicands a = ( + 7-6) 2 Distributive Property b = 6 2 Simplify = (10 + ) 7 + ( - 6) 11 Distributive Property = Simplify. 1A B C D Not all radical expressions have like radicands. Simplifying the expressions may make it possible to have like radicands so that they can be added or subtracted. 619

2 Study Tip Simplify First Simplify each radical term first. Then perform the operations needed. Example 2 Add and Subtract Expressions with Unlike Radicands Simplify = 2( 2 2 ) + 2( 2 2 ) + ( ) Product Property = 2( 2 ) + 2( 2 ) + (6 2 ) Simplify. = Multiply. = 20 2 Simplify. 2A B C D Multiply Radical Expressions Multiplying radical expressions is similar to 2 multiplying monomial algebraic expressions. Let x 0. Monomials (2x)(x) = 2 x x Radical Expressions (2 x )( x ) = 2 x x = 6 x 2 = 6x You can also apply the Distributive Property to radical expressions. Example Multiply Radical Expressions Watch Out! Multiplying Radicands Make sure that you multiply the radicands when multiplying radical expressions. A common mistake is to add the radicands rather than multiply. a = ( 2) ( 2 6 ) Associative Property = 6( 12 ) Multiply. = 6(2 ) Simplify. = 12 Multiply. b. (2 + ) (2 + ) = ( 2 ) + ( ) Distributive Property = [( 2)( )] + [( )( )] Associative Property = [6( 2 )] + [1( 1 )] Multiply. = [6()] + [1( 1 )] Simplify. = Multiply. A B C. 2 ( ) D. ( 2 - ) You can also multiply radical expressions with more than one term in each factor. This is similar to multiplying two algebraic binomials with variables. 620 Lesson 10- Operations with Radical Expressions

3 Review Vocabulary Factored Form A monomial is in factored form when it is expressed as the product of prime numbers and variables. (Lesson 8-1) Real-World Example Multiply Radical Expressions GEOMETRY Find the area of the rectangle in simplest form. A = ( 2 - )( + ) A = l w First Terms Outer Terms Inner Terms Last Terms 2 - = ( 2 )( ) + ( 2 )( ) + (- )( ) + ( )( ) = Multiply. = Simplify.. GEOMETRY The area A of a rhombus can be found using the equation A = 1_ 2 d 1 d 2, where d 1 and d 2 are the lengths of the diagonals. What is the area of the rhombus at the right? Concept Summary Operations with Radical Expressions Operation Symbols Example addition, b 0 subtraction, b 0 multiplication, b 0, g 0 a b + c b = (a + c) b like radicands a b + c b = (a - c) b like radicands a b (f g ) = af bg Radicands do not have to be like radicands. + 6 = ( + 6) = = (12-8) = 2 ( 7 ) = ( ) ( 2 7 ) = 1 1 Check Your Understanding = Step-by-Step Solutions begin on page R12. Examples ( 6 ) 10. (8 ) 11. ( ) 12. ( ) Example 1. GEOMETRY The area A of a triangle can be found by using the formula A = 1_ bh, where 2 b represents the base and h is the height. What is the area of the triangle at the right?

4 Practice and Problem Solving Extra Practice begins on page 81. Examples ( ) ( ) 21. ( ) 22. ( ) 2. ( - 2 )( ) 2. ( )( ) 2. ( 2 + )( ) Example B 26. GEOMETRY Find the perimeter and area of a rectangle with a width of and a length of _ _ _ _ _ 1. ( - ) 2 2. ( 2 + ) 2 ROLLER COASTERS The velocity v in feet per second of a roller coaster at the bottom of a hill is related to the vertical drop h in feet and the velocity v 0 of the coaster at the top of the hill by the formula v 0 = v 2-6h. a. What velocity must a coaster have at the top of a 22-foot hill to achieve a velocity of 120 feet per second at the bottom? b. Explain why v 0 = v - 8 h is not equivalent to the formula given.. FINANCIAL LITERACY Tadi invests $22 in a savings account. In two years, Tadi has $270 in his account. You can use the formula r = _ v 2-1 to find the average annual interest rate r that the account has earned. The initial investment is v 0, and v 2 is the amount in two years. What was the average annual interest rate that Tadi s account earned?. ELECTRICITY Electricians can calculate the electrical current in amps A by using the formula A = _ w, where w is the power in watts and r the resistance in ohms. How r much electrical current is running through a microwave oven that has 80 watts of power and ohms of resistance? Write the number of amps in simplest radical form, and then estimate the amount of current to the nearest tenth. v 0 H.O.T. Problems Use Higher-Order Thinking Skills C 6. CHALLENGE Determine whether the following statement is true or false. Provide an example or counterexample to support your answer. x + y > x 2 + y 2 when x > 0 and y > 0 7. REASONING Let a, b, c, d, and f be rational numbers. Show that if you multiply a b + c f and a b - c f, the product has no radicals. Explain why this occurs. 8. OPEN ENDED Write an equation that shows a sum of two radicals with different radicands. Explain how you could combine these terms. 9. WRITING IN MATH Describe step by step how to multiply two radical expressions, each with two terms. Write an example to demonstrate your description. 622 Lesson 10- Operations with Radical Expressions

5 Virginia SOL Practice A.1 0. SHORT RESPONSE The population of a town is 1,000 and is increasing by about 20 people per year. This can be represented by the equation p = 1, y, where y is the number of years from now and p represents the population. In how many years will the population of the town be 1,00? 1. GEOMETRY Which expression represents the sum of the lengths of the 12 edges on this rectangular solid? A 2(a + b + c) B (a + b + c) C (a + b + c) D 12(a + b + c) a b c 2. Evaluate n - 9 and n - 9 for n = 2. F ; G ; 2 H 2; J 2; 2. The current I in a simple electrical circuit is given by the formula I = _ V, where V is the R voltage and R is the resistance of the circuit. If the voltage remains unchanged, what effect will doubling the resistance of the circuit have on the current? A The current will remain the same. B The current will double its previous value. C The current will be half its previous value. D The current will be two units more than its previous value. Spiral Review Simplify. (Lesson 10-2) a b x y c d f Graph each function. Compare to the parent graph. State the domain and range. (Lesson 10-1) 0. y = 2 x 1. y = - x 2. y = x + 1. y = x -. y = x +. y = x FINANCIAL LITERACY Determine the value of an investment if $00 is invested at an interest rate of 7.2% compounded quarterly for 7 years. (Lesson 9-7) Factor each trinomial. (Lesson 8-) 7. x x y 2 + 1y p 2-17p x 2 + 6x y 2 - y w + w 2 Skills Review Solve each equation. Round each solution to the nearest tenth, if necessary. (Lesson 2-) 6. -c = q -.7 = m + = _ n = _-h - (-) = t t =

6 Algebra Lab Simplifying n th Root Expressions The inverse of raising a number to the n th power is finding the n th root of a number. The index of a radical expression indicates to what root the value under the radicand is being taken. The fourth root of a number is indicated with an index of. When simplifying a radical expression in which there is a variable with an exponent in the radicand, divide the exponent by the index. quotient 1 = 2 R index x 1 = x 2 x remainder Example 1 a. x 7 x 7 = x 2 x 7 = 2 R 1 b. 2 x 9 2 x 9 = 2 x 9 Multiplication Property = 2x x 9 = 1 R The properties of square roots (and nth roots) also apply when the radicand contains fractions. Separate the numerator and denominator and then simplify them individually. Example 2 _ Simplify 27 x 8 y. _ 27 x 8 y = _ 27 _ x 8 y Multiplication Property of Radicals = _ 2 _ x x 2 y Simplify. = _ x x 2 2y Multiplication Property of Radicals The indices and the radicands must be alike in order to add or subtract radical expressions. Example Simplify 8 _ + _ - _ + _. Combine the expressions with identical indices and radicands. Then simplify. 8 _ + _ - _ + _ = (8 ) _ + _ + _ = _ + _ + _ Associative Property Simplify. 62A Extend 10- Algebra Lab: Simplifying nth Root Expressions

7 When multiplying radical expressions, ensure that the indices are the same. Then multiply the radicands and simplify if possible. Once none of the remaining terms can be combined or simplified, the expression is considered simplified. Example Simplify Multiply the radicands with identical indexes = ( 2)( 6 12 ) 10 Associative Property = 10 ( 6 12 ) 10 Multiply. = Multiply. The properties of radical expressions still hold when variables are in the radicand. Example Simplify 6 x x + ( x + 2 x ). Follow the order of operations and the properties of radical expressions. 6 x x + ( x + 2 x ) = 6 x x + ( x ) = 6 x x + ( x ) = 6 x + 9 x = 6x + 9 x Add like terms. Associative Property Multiply. Simplify. Exercises 1. c _ 8 a 12 b d 9. _ 2 x y 6 z x x x _ 2 + _ 2 2 2_ a 2 a 2( a + a ) 10. x_ + x_ 2 2x _ 11. _ 8 a 2 1 b _ 2 a 27b 12. _ 16 x 81 y + _ x y + _ 16x y 8 Think About It 1. Provide an example in which two radical expressions with unlike radicands can be combined by addition. 1. Provide an example in which two radical expressions with identical indices and with like variables in the radicand cannot be combined. 62B