Chapter 9. Worked-Out Solutions. Check. Chapter 9 Maintaining Mathematical Proficiency (p. 477) y = 2 x 2. y = 1. = (x + 5) 2 4 =?

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1 Maintaining Mathematical Proficienc (p. ). x + 0x + x + (x)() + (x + ). x 0x + 00 x (x)(0) + 0 (x 0). x + x + x + (x)() + (x + ). x x + x (x)(9) + 9 (x 9). x + x + x + (x)() + (x + ) Check x x +? ()? () +?? + The solution is (, ). 9. x + x x + (, ). x 0x + x (x)() + x (x ) x. x + x x + x x (, ) The lines appear to intersect at (, ). Check x + x? () +? ()? +? The solution is (, ).. x x + (, ) x + x x The lines appear to intersect at (, ). The lines appear to intersect at (, ). Check x + x? ( ) +? ( )? +? The solution is (, ). 0. A polnomial of the form x + bx + c is a perfect square trinomial when b is twice the square root of c. So, the value of c must be ( b ). Mathematical Practices (p. ). Sample answer: Guess Check How to Revise. (.). 0.0 Decrease guess.. (.). 0.0 Increase guess.. (.) Increase guess... (.) (.) (.) Decrease guess. Decrease guess. The solution is between. and.. So, to the nearest thousandth, the negative solution of the equation is x.. Copright Big Ideas Learning, LLC Algebra

2 . Sample answer: Guess Check How to Revise. (.) Increase guess.. (.) Decrease guess..0.0 (.0) (.0) (.0) (.0) Decrease guess. Increase guess. Increase guess. The solution is between.0 and.0. So, to the nearest thousandth, the positive solution of the equation is x.0. Guess Check How to Revise. (.). 0.0 Decrease guess (.). 0.0 (.0) (.0) (.0) (.0) Increase guess. Increase guess. Decrease guess. Decrease guess. The solution is between.0 and.0. So, to the nearest thousandth, the negative solution of the equation is x Explorations (p. 9). a. + + and Because 0, + does not equal +. So, the general expressions a + b and a + b are not equal. b. 9 and 9. Because, 9 9 is true. Also, a b a / b /, and b the Power of a Product Propert, a / b / (a b) /. Also, (a b) / a b. So, the general expressions a b and a b are equal. c. and. Because, does not equal. So, the general expressions a b and a b are not equal. d and 00. Because, 00 a is true. Also, b of a Product Propert, a/ b / ( a So, the general expressions a b a/, and b the Power b/ b) /. Also, ( a b) / a and a are equal. b. Sample answer: A counterexample for adding square roots is 9 +, and a counterexample for subtracting square roots is 9.. Multipl or divide the numbers inside the square root smbols and take the square root of the product or quotient.. Sample answer: An example of multipling square roots is 9 9. An example of dividing square roots is.. a. Because a b and a b are equal, an algebraic rule for the product of square roots is a b a b. and a are equal, an algebraic rule for b b. Because a b the quotient of square roots is a b 9. Monitoring Progress (pp. 0 ) a b x 9 x x 9 x x x x. n n n n n n n n n n n z z z x. x x x x b. Algebra Copright Big Ideas Learning, LLC

3 9. 0. x x x x x x x x x x x. a a a a c. d c d c d c c d c c d c d c c d c.. x x x x x x x x x x ( ) ( ) () ( + ) ( ) + () ( ) ( ) ( ) Copright Big Ideas Learning, LLC Algebra 9

4 . d h () You can see 0, or about. miles.. Let be the length of the longer side ( + ) 0 ( + ) ( + ) () The length of the longer side is about feet ( + 0). 9 9 ( ). x x ( ) x x. ( + ) ( + ) ( + ) ( + ) ( + ) [ ( + ) ] ( ). ( ) ( ) ( ) () + ( ) ( ) () + 9. Exercises (pp. ) Vocabular and Core Concept Check. The process of eliminating a radical from the denominator of a radical expression is called rationalizing the denominator.. The conjugate is. 9. First, rewrite x 9 as x. Then, b the Product 9 x Propert of Square Roots, 9 x. Also, 9 9. So, x and x are equivalent. 9. The expression that does not belong is. The other three expressions have like radicals of. Monitoring Progress and Modeling with Mathematics. The expression 9 is in simplest form.. The expression is not in simplest form because the radicand is a fraction.. The expression is not in simplest form because the radicand has a perfect square factor of.. The expression is in simplest form. 9. The expression is not in simplest form because a radical appears in the denominator of the fraction. 0. The expression 0 is in simplest form.. The expression is not in simplest form because a + radical appears in the denominator of the fraction.. The expression is not in simplest form because the radicand has a perfect cube factor of. 0 Algebra Copright Big Ideas Learning, LLC

5 b b. x x b x b x 9. m m m m m 9m m 0. n n n n n n n n n n n a. 9 a 9 a a a a a a. 00 x 00 x 0 x 0 x x. 9.. k k k v. v v v x x x x x x x. n n n n c. c c c. 000x 000x 000 x 000 x 0x. a b a b a b a b ab ab h h. h h h h h h. The radicand has a perfect square factor of 9. So, it is not in simplest form. Copright Big Ideas Learning, LLC Algebra

6 . The denominator should be. 9. To rationalize the denominator of the expression, multipl b a factor of. 0. To rationalize the denominator of the expression, z multipl b a factor of z. z. To rationalize the denominator of the expression, x x multipl b a factor of. x. To rationalize the denominator of the expression m, multipl b a factor of.. To rationalize the denominator of the expression, multipl b a factor of To rationalize the denominator of the expression, + multipl b a factor of a a a a a a a a d. d d d d d x x x x x x x x. n x x x x n n n n n n n n n n 9n n n 9 n n n n n n Algebra Copright Big Ideas Learning, LLC

7 ( ) ( ). + + ( + ) ( ) ( + ) ( + ) ( + ) ( ) ( ) ( ) Copright Big Ideas Learning, LLC Algebra ( + ) ( ) ( ) ( + ) ( + ) ( ) ( ) ( ) 9 h. a. t It takes, or about. seconds for the earring to hit the ground. b. h h t. The earring hits the ground about.. 0. second sooner when it is dropped from two stories below the roof.. a. P d d d d d d d So, the formula for a planet s orbital period is P d d. b. P d d...(.0). Earth ears

8 . I P R 9 9 The current the appliance uses is, or about. amperes. v. Account : r v % v Account : r v 0 Account : r v v Account : r v % v % % v Account : r % v 0 Invest mone in Account because it has the greatest interest rate of about.%.. h(x) x. g(x) x h(0) (0) g(0) (0) 0 0 So, h(0), or So, g(0), or about.0. about... r(x) x x + r() () () + () So, r(). p(x) x x p() () , or So, r() 0 0, or about 0.. about a + bc ( ) + () ( ) +, or about. 0. c ab ( ) ( )() , or about a + b ( ) + () + +, or about.9. b ac ( ) ( ) +, or about. 0 0 Algebra Copright Big Ideas Learning, LLC

9 .. + w w ( + ) w ( + ) ( + ) ( + ) w + + ( ) ( ) ( + ) +. The width of the text is about. inches. + w w ( + ) w ( + ) ( + ) ( + ) w + + ( ) ( ) ( + ) +.9 The width of the flag is about.9 inches ( + ) +. ( ). 9 9 ( ) ( + ) ( + ) ( + ) ( + ). t t t t t t t t t t ( ) t t. ( + ) ( + ) 0 0 Copright Big Ideas Learning, LLC Algebra

10 . ( ) ( ). ( x 9x ) x 9x x 0x 0x 0x 0x 0x 0x 0x ( ) 0x 0x. ( + ) ( + 0). ( 9 ) ( ) ( )( 9 ) + ( 9 ) ( ) ( + ) ( 0 ) ( + ) 9. ( + ) ( + ) 90. ( ) 9. C π a + b π 0 + π 00 + π π π π π π ( ) The circumference of the room is about square feet. 9. a. The expression + represents an irrational number because is not a perfect square. b. The expression represents a rational number. B the Quotient Propert of Square Roots,. is equal to, and is a perfect square. So,, and is a rational number. c. The expression represents an irrational number because is not a perfect square. d. The expression + represents an irrational number because and are not pefect squares. a e. The expression represents an irrational number 0 because and 0 are not perfect squares. + f. The expression represents a rational number. b + b B the Distributive Propert, + + b +, b b ( + ) and when ou simplif the expression, ou get, which is b a rational number when b is a positive integer. Algebra Copright Big Ideas Learning, LLC

11 9. x x x x x x x x x x x x x x x x x x x ( + ) ( + ) a π + + π + + π 0 0 π π + 0 π π + π + π π π + π π b. 0 π 0 π 0 π π 0 π π π π 0 π π π 00. a. The sum of a rational number and a rational number is alwas rational because the sum of two fractions can alwas be written as a fraction. b. The sum of a rational number and an irrational number is alwas irrational because if one of the factors is a nonrepeating decimal, then the sum cannot be written as the ratio of two integers. c. The sum of an irrational number and an irrational number is sometimes irrational. The sum is either 0, or it is irrational. For example, +, which is irrational. However, + ( ) 0, and zero is a rational number because it can be written as a ratio of two integers, such as 0 0. d. The product of a rational number and a rational number is alwas rational because the product of two fractions can alwas be written as a fraction. e. The product of a nonzero rational number and an irrational number is alwas irrational because if one of the factors is a nonrepeating decimal, then the product cannot be written as the ratio of two integers. f. The product of an irrational number and an irrational number is sometimes irrational. An example of a product that is irrational is π π, but an example of a product that is rational is. 0. The simplified form of the expression m contains a radical when m is odd, because to an odd power is not a perfect square. The simplified form of the expression m does not contain a radical when m is even, because to an even power is a perfect square. 0. Sample answer: If s, then the side length,, is an irrational number, the surface area is [ ( ) ], which is an irrational number, but the volume is ( ), which is a rational number. 0. When a < b, if ou multipl each side of the inequalit b a, ou get a < ab. Similarl, when a < b, if ou multipl each side of the inequalit b b, ou get ab < b. So, putting these two inequalities together, ou get a < ab < b. When ou take the square root of each part of this inequalit, ou get a < ab < b. So, it must be that ab lies between a and b on a number line. Copright Big Ideas Learning, LLC Algebra

12 0. Your friend is incorrect. Using the sum and difference pattern to simplif the product of the denominator + and, ou get ( ), which means the denominator will still contain a radical x x ( + ) 0 x ( + ) 0 ( + ) ( + ) 0 x ( ) ( ) 0 ( ) 0 ( ) 0 ( ) 0 0 ( ) So, the preceding term is. 0. a. x x 0 ( + ) ( + )? 0 + () ( ) + ( ) + ( + )? ? 0 + +? ( )? ( )? ( )? 0 +? 0 ( ) + ( )? ? ( ) x x 0 ( )? 0 () ( ) + ( ) + ( )? ( + )? 0 + +? 0 + ( + )? 0 + ( + )? 0 + ( ) +? 0 +? 0 ( ) + ( + )? 0 b. Sample answer: DF + A D E C F B 0 + 0? In order to rationalize the denominator of x +, let a x and let b and multipl the numerator and denominator b a ab + b ( x ) x () + x x +. x x + x + x x + ( x x + ) ( x ) + x x + x + So, x + x x +. x + Maintaining Mathematical Proficienc 0. To graph x, use slope m and -intercept b. The graph crosses the x-axis at (, 0). So, the x-intercept is. x x Algebra Copright Big Ideas Learning, LLC

13 09. To graph x +, use slope m and -intercept b. x + x The graph crosses the x-axis at (, 0). So, the x-intercept is. 0. To graph x, use slope m and -intercept b. x x The graph crosses the x-axis at (, 0). So, the x-intercept is.. To graph x +, use slope m and -intercept b. x +. ( ) x x Check ( ( ) x ( ) x x ( x) x ( x) x () (x) x x +x +x x The solution is x.. x ( ) x + ( ) x ( ) x + x ( ) x + x (x + ) x (x + ) x (x) () x x +x +x x x x The solution is x. 9. Explorations (p. 9). a. x ) x x ( )? ( )??,, Check x ( ) x + /? ( ) / + ( ) /? ( ) / ( ) /? [ ( ) ] / ( 90,) /? ( ) / ( 90,) / ( 90,) / x The graph crosses the x-axis at (, 0). So, the x-intercept is.. x Check x x? x The solution is x. 0 9 x x. x x Check x x ( ) x x x x x x x x x x x The solution is x.?? 9 9,? 9 9, 9, Copright Big Ideas Learning, LLC Algebra 9 x b. An x-intercept of a graph is the x-coordinate of a point where the graph crosses the x-axis. This graph crosses the x-axis at two points. So, it has two x-intercepts. The are 0 and.

Properties of Radicals

9. Properties of Radicals Essential Question How can you multiply and divide square roots? Operations with Square Roots Work with a partner. For each operation with square roots, compare the results obtained