Cube Root Equations VOCABULARY


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1 104 Cube Root Equations TEKS FOCUS TEKS (6)(B) Solve cube root equations that have real roots. TEKS (1)(B) Use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution. Additional TEKS (1)(A), (1)(C), (1)(D), (7)(H) VOCABULARY Cube root equation A cube root equation is a radical equation in which the radical has an index of. A cube root equation can also be written using a rational exponent with a denominator of. Formulate create with careful effort and purpose. You can formulate a plan or strategy to solve a problem. Reasonableness the quality of being within the realm of common sense or sound reasoning. The reasonableness of a solution is whether or not the solution makes sense. Strategy a plan or method for solving a problem ESSENTIAL UNDERSTANDING Solving a cube root equation may require that you cube each side of the equation. Problem 1 TEKS Process Standard (1)(B) Solving a Cube Root Equation With Real Roots What is the solution to the equation 1 5x 1 = 4? Evaluate your problemsolving process. Analyze Given Information The cube root needs to be removed to solve for x. What property allows you to cube each side of the equation? If a = b, then a n = b n for any integer n. Formulate a Plan Solve for x algebraically. 1. To do so, first remove the cube root by cubing each side of the equation.. Solve the resulting linear equation by simplifying and combining like terms.. As a final step, isolate x by dividing by 5. Determine a Solution Execute the plan to solve for x. 1 5x  1 = 4 (1 5x  1) = (4) Cube each side. 5x  1 = 64 Simplify. 5x = 65 Add 1 to each side. x = 1 Divide each side by 5. The solution is 1. continued on next page 46 Lesson 104 Cube Root Equations
2 Problem 1 continued Justify the Solution Check your solution in the original equation. Problem 1 5x  1 = 4 Write the original equation. 1 5(1) Substitute 1 for x Simplify = 4 Evaluate the ProblemSolving Process The solution checks, since 4 = 4. The problemsolving process was successful. Solving Equations With Rational Exponents A What are the solutions of (x + 1) = 1? An equation in which the Solutions of the equation reciprocal power. Why is this a cube root equation? You can rewrite the equation using radical notation as (x + 1) = 4. (x + 1) = (x + 1) = 4 Divide each side by. ((x + 1) ) = 4 (x + 1) = 4 Raise each side to the power. Use absolute value symbols because the denominator x + 1 = 8 in the exponent indicates square root. x + 1 = {8 x = 7 or x =9 The solutions are 7 and 9. Check (x + 1) = 1 (x + 1) = 1 (7 + 1) 1 (9 + 1) 1 ( ) 1 (() ) 1 () 1 () 1 1 = 1 1 = 1 continued on next page PearsonTEXAS.com 47
3 Problem continued Why do you isolate the variable expression? If you raise each side of (x + 1) = 97 to the 5 power you will end up with a more complicated equation, not a simpler one. B What is the solution of (x + 1) = 97? The solution is 7. (x + 1) = 97 (x + 1) = 97 Rewrite the radical using a rational exponent. (x + 1) 5 = 96 Subtract 1 from each side. (x + 1) 5 = Divide each side by. 5 ((x + 1) ) 5 5 = Raise each side to the 5 power. x + 1 = 8 Simplify. x = 7 Subtract 1 from each side. Problem TEKS Process Standard (1)(A) Using a Cube Root Equation Earth Science For Meteor Crater in Arizona, the formula d = 50. V relates the diameter d of the rim (in meters) to the volume V (in cubic meters). What is the volume of Meteor Crater? (All values are approximate.) 1. km What is the diameter in meters? 1. km = 1. * 1000 m d = 50. V d = 5 0. V ( d ) = V 0. Solve for V. First divide each side by. Cube each side. 0.( d ) = V Multiply each side by ( 100 ) = V Substitute 100 for d. 64,800,000 = V Simplify. The volume of Meteor Crater is about 64,800,000 m. 48 Lesson 104 Cube Root Equations
4 Problem 4 TEKS Process Standard (1)(C) Solving a Cube Root Equation by Graphing Multiple Choice You can model the population P of Corpus Christi, Texas, between the years 1970 and 005 by the cube root function P (x) = 75,000 1x 1950, where x is the year. Using this model, in what year was the population of Corpus Christi 50,000? How can you rewrite a cube root function using an exponent? You can write a cube root function y = 1x as y = x For P = 50,000, solve the equation 50,000 = 75,000 1x Graph Y 1 = 75000(X 1950)^(1/) and Y = Adjust the window to find where the graphs intersect. Use the INTERSECT feature to find the xcoordinate of the intersection. In the year 1987, the population of Corpus Christi was 50,000. The correct answer is C. Intersection X= Y=50000 ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. Solve each equation. Check your answer and evaluate your problemsolving process. For additional support when completing your homework, go to PearsonTEXAS.com x + 7 = x =. 1 = + 1 4x x + = x + = x  1 = 7. 1 x +  = x = x  = Apply Mathematics (1)(A) A diameter of a spherical water tank is 6 ft. What is the volume of the tank? 1Hint: d = 6V 5 p 11. The formula d = 4V 5 p gives the diameter of a closed cylinder where V is the volume. Boyle s Law says that P initial V initial = P new V new. If the pressure P initial is 1 psi, P new is 8 psi and V initial is p, what is the new diameter using Boyle s Law? 1. Use a ProblemSolving Model (1)(B) The x function y = 5 4p relates the radius y of a spherical gas tank (in meters) to the volume x (in cubic meters). A company manufactures gas tanks with a radius of 10 m. Find the volume of the tank to the nearest cubic meter. Use a problemsolving model by analyzing the given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process. PearsonTEXAS.com 49
5 Solve each equation. 1. (x + 5) = (x + ) = (x  ) 4 = (4  x) = (x) = (x + )  = (x  1) = 6 0. (x  ) = x  7 Select Tools to Solve Problems (1)(C) Solve each cube root equation by graphing. Round the answer to the nearest hundredth, if necessary. If there is no solution, explain why x  = 1. 1 x  = 4. 1 x + 5 = 1  x 4. 1 x = 1(x + 1) 5. 1(x + ) = 41 (x) x  1 = 1x Suppose that a function pairs elements from set A with elements from set B. Recall that a function is called onto if every element in B is paired with at least one element in A. a. The graph shows a transformation of y = 1 x. Write the function. b. What are the domain and range of the function? c. If the domain is restricted to all real numbers greater than or equal to 10, and the range is the set of nonnegative real numbers, is the function onto? Explain. 8. The ninebanded armadillo is a relatively recent addition to Texas. It can jump 4 ft and grows to about 15 in. from the neck to the base of the tail. Some armadillos roll up into a ball when frightened. The spherical shape can be used to show that the function y = 1 6p x relates the length y of the armadillo (in inches) to its volume x (in cubic inches). Suppose you measure the length of an armadillo as 18 in. Write and solve a cube root equation to find the volume of the armadillo to the nearest cubic inch y x O Lesson 104 Cube Root Equations
6 9. Explain how you would find the x and yintercepts of f (x) = 1 x +. Then find the intercepts and graph the function. 0. The size of a computer case is related to the size of the motherboard, and smaller cases mean that upgrading is limited. The equation s = 1 V models the length of an edge of a computer case with volume V in cubic inches. a. Graph the equation on your calculator. b. Suppose you want to buy a new video card for your old case that has volume 51 in.. You need 0.75 inch minimum between the case and the edge of the video card for air circulation. If video cards come in fulllength 1 inches, halflength 7 inches, and lowprofile 6.5 inches, which one would be the best choice? Explain. (Video cards are installed at right angles to the sides of the case.) TEXAS Test Practice 1. How is the graph of y = 1 x  5 translated from the graph of y = 1 x? A. shifted 5 units left B. shifted 5 units right C. shifted 5 units up D. shifted 5 units down. Which absolute value inequality has the graph shown here? F. 0 x G. 0 x Ú H. 0 x J. 0 x Ú. Which polynomial cannot be factored in the real number system? A. x  x + B. x + 4 C. 4x  1 D. x y  xy 4. How do the domains and ranges of f (x) = 1x  1 and g(x) = 1x  1 compare? PearsonTEXAS.com 441
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