Section 9.1 Roots, Radicals, and Rational Exponents

Transcription

1 Section 9. Roots, Radicals, and Rational Exponents SQUARE ROOTS The square root of a is written as N. If N ;, then b a. Note: For to be defined in the real number system, a 0. NOTATION: a / a is the RADICAL NOTATION for the square root of a. is the RATIONAL EXPONENT NOTATION for the square root of a. EXAMPLES: a) because b) a / / c) ( ) is not a real number because there is no number, squared, that will give THE n-th ROOT The nth root of a is radicand. n a where n is the index or root number anda is called the Note: For to be defined in the real number system, if n is even, then N 0. NOTATION: n a n a a / n is the RADICAL NOTATION for the nth root of a. a /n is the RATIONAL EXPONENT NOTATION for the nth root of a. EXAMPLES: / a) Calculator Entry: ^(/) 7 / 7 b) 87 ( 87) Calculator Entry: ( 87)^(/7).7 c) Calculator Entry: ^(/) / d) ( ) is not a real number Calculator Entry: (-)^(/) 8

2 RATIONAL EXPONENTS EXAMPLES: a) 8 / é 8 7 b) / d ( ) c) / $ & % # " m n n m a a ( ) n a m Problem YOU TRY Rewriting Radical/Rational Exponents Complete the table below. Each expression written in radical notation should be written with rational exponents and vice versa. Assume all variables are positive. Written in radical notation Work Written with rational exponents # # # # (# ) # # / # a) c) A é A c d) b e) 0 / f) / g) # S/ h) X S/ 9

3 Problem YOU TRY Compute with Radical/Rational Exponents Complete the table below. Each expression written in radical notation should be written with rational exponents and vice versa. Then evaluate using the calculator. Written in radical notation Written with rational exponents Evaluate using calculator (Rounded to two decimal places) ^(/).7 a) 7 d b) c) d d) ( ) e) f) S g) 8 h) ( 9) 0

4 Problem MEDIA/CLASS EXAMPLE Compute with Radical/Rational Exponents Compute each of the following WITHOUT a calculator showing as much work as possible. 9 8 a) b) 9 8 c) d) e) f) ( ) Problem YOU TRY Compute with Radical/Rational Exponents Compute using a calculator. (Round answers to decimal places) 7 9 a) b) & 9 8 c) $ 9 " # d) %

5 Problem YOU TRY Compute with Radical/Rational Exponents Compute each of the following WITHOUT a calculator showing as much work as possible. a) b) c) / d) ( ) e) f) ( 7) / ( ) g) h) $ & 7 " # % Problem YOU TRY Compute Radicals Complete the table without the using a calculator. a) " # # + b) # # x " # # + 0 x # # 8 8

6 c) h # # d) 9 # # x h # # 0 x 9 # # 0 9 8

7 Section 9. Operations with Radical Expressions Operations on radical expressions work the same way as operations on polynomial expressions. When multiplying two monomial expressions, you multiply like factors, that is, you multiply coefficients together and variables together. Similarly, when multiplying two monomial radicals, you multiply the numbers outside the radical sign together and the radicands together, M M as long as the radicals have the same index. To multiply radicands, we use the rule: N ; M N ;. (Note: When n is even, this rule works only for positive radicands.) Problem 7 WORKED EXAMPLE Multiplying Radicals Multiply the following. Be sure to simplify all answers. a) Multiply the two radicands using Simplify b) Multiply the two radicands using Simplify 8 Problem 8 YOU TRY Multiplying Radicals Multiply the following. Be sure to simplify all answers. d a) b) d

8 When adding or subtracting polynomial expressions, you add and subtract coefficients whose variable and its exponent are alike. Similarly, when adding or subtracting radical expression, you add and subtract the number outside the radical sign as long as the radical expression s index and radicand are alike. Problem 9 WORKED EXAMPLE Multiplying Radicals Use the distributive property to multiply the following radicals. Simplify all answers. a) 7 b) ( ) ( 7) ( + 8)( ) ( + 8)( ) c) d) ( 0 + ) ( 8 7)( 8 + 7) ( 0 + ) ( 0 + )( 0 + ) ( 8 7)( 8 + 7)

9 Problem 0 MEDIA/CLASS EXAMPLE Multiplying Radicals Use the distributive property to multiply the following radicals. Simplify all answers. a) 8 ( + ) b) ( 7 + )( ) c) d) + ( ) + ( ) ( ) Problem YOU TRY Multiplying Radicals Multiply the following. Be sure to simplify your answers. a) b) + 8 ( ) ( ) ( + ) ( ) c) 8 7 d)

10 Problem YOU TRY Multiplying Radicals Multiply the following. Be sure to simplify your answers. a) 7 b) ( + )( ) ( 8 )( + 8) ( )( ) ( 9 + )( 9) c) 0 d) ( + ) ( ) e) f) ( + )( ) ( 0 + 8) ( 0 8) g) 9 9 h) 7

11 Section 9. Solving Radical Equations To solve radical equations algebraically: Isolate the radical part of the equation on one side and anything else on the other Sometimes you will have radicals on both sides. That is ok. Raise both sides of the equation to a power that will undo the radical ( nd power to get rid of square root, rd power to get rid of cube root, etc ) Solve. Check your answer Not all solutions obtained will check properly in your equation. Problem WORKED EXAMPLE Solve Radical Equations Algebraically Solve the equation 0 x algebraically. First, square both sides to remove the square root. 0 x ( 0 x ) 0 x Next, isolate x. VERY IMPORTANT Check x in the original equation to be sure it works Not all solutions obtained using the process above will check properly in your equation. If an x does not check, then it is not a solution. 0 ( ) x 0 x x x is the solution to this equation