Math 2 UNIT 3: Radicals

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Bethany Wheeler
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1 Math 2 UNIT : Radicals 1

2 Date Lesson Quiz Topic Homework Monday 2/20 Tuesday 2/21 Wednesday 2/22 Simplifying Radicals Solving Radical Equations Solving Radical Equations Thursday 2/2 Exponential Rules Quiz on Days 1 2 Friday 2/24 Solving with Rational Exponents Monday 2/27 Radical Applications Tuesday 2/28 Wednesday /1 Thursday /2 ACT Practice Day Review Unit Test State Standards: NC.M2.N-RN.1 Explain how expressions with rational expressions can be rewritten as radical expressions. NC.M2.N-RN.2 Rewrite expressions with radicals and rational exponents into equivalent expressions using the properties of exponents. NC.M2.A-SSE.1 Interpret expressions that represent a quantity in terms of its context. a. Identify and interpret parts of a quadratic, square root, inverse variation, or right triangle trigonometric expression, including terms, factors, coefficients, radicands, and exponents. b. Interpret quadratic and square root expressions made of multiple parts as a combination of single entities to give meaning in terms of a context. NC.M2.A-CED.1 Create equations and inequalities in one variable that represent quadratic, square root, inverse variation, and right triangle trigonometric relationships and use them to solve problems. NC.M2.A-REI.1 Justify a chosen solution method and each step of the solving process for quadratic, square root and inverse variation equations using mathematical reasoning. NC.M2.A-REI.2 Solve and interpret one variable inverse variation and square root equations arising from a context, and explain how extraneous solutions may be produced. 2

3 Day 1: Simplifying Radicals Radicals: n x = r ***If there is no index, it is understood to be. 64 = 4 (because 4*4*4=64) index= radicand= root= Rewrite the expression using radical notations: x f n f n = x is the index and is the exponent (power). Examples: / 2. 4y 1/5. (2x) 2/ You Try: 4. /4 5. X 1/7 6. (y) 4/5 Simplifying Radical Expressions: Examples: y m 4 n v 8

4 Practice simplify the following radicals: x x n x x x y 6 4

5 Day 1 Homework: Simplify the following n r a b x 7 y x y x 9. 56x 5 y 5

6 Day 2: Solving Radical Equations Warm-Up Simplify x 7 2. Write in radical form x /2. Solve for x 5x + 18 = 58 Steps for Solving Radical Equations: 1) Isolate the radical. 2) Undo the radical by raising it to the nth root (n=index). ) Isolate the variable and solve. 4) Check for Extraneous solutions Examples: 1. 4x 8 = 0 2. x + 6 = x 6

7 Solving a Radical Equation Activity 1. Cut out the boxes from the given handout. 2. Place the cutouts in order and glue it below, beginning with the original equation.. Write the steps to solving the equation to the side of your cut outs. 7

8 Practice solving radical equations: 1. 5 x + 2 = x + 1 = 8. 2x 4 = x + 1 = 2x x 2 x = 1 6. x + 4 = 8

9 Day 2 Homework: Solve for x 1. x x = 2. x 2 = 5. x 2 = x 5 = a = x = 0 9

10 7. 2x + 1 = 8 8. x 2 = x 6 5x + 2 = x 2 = 8 10

11 Day : Solving Radical Equation cont. Warm Up Simplify the following: x 5 2. x 2 y 2058x y 2 Solve the equations: 4. x 9 = 4. x = 0 11

12 Day : Simplifying and Solving Radicals Practice x + 6 = x + 7 = x 5 4 = x x 6 y x 8 y z 5 12

13 10. x 6 4 = x = x = x + 4 = x x 9 y 7 1

14 Day Homework: 14

15 Day 4: Exponents Converting from Radicals to Rational Exponents: n f f x = x n Examples: t = 2. 5x = You Try. 2x = Review of Exponent Rules: Multiplying Monomials with like bases 2 4 = y 4 y 10 = x x 14 = When multiplying monomials with like bases, we the exponents. Raising a power to a power ( 2 ) 6 = (x ) 5 = (2x 4 ) = When raising a power to a power, we the exponents. Dividing monomials with like bases 6 x 2 = 8 y x = 5 y 12 = When dividing monomials with like bases, we the exponents. **We cannot leave anything with a negative exponent, so if the exponent is negative, it or put it under 1 and change the exponent to a. 15

16 Examples: 1. (x 5 y 8 ) 2 2. ( 6x 8 y ) 2. ( ( x 5 ) (12x ) )2 Practice: Convert from Radicals to Rational Exponents: 16

17 17

18 Day 4 Homework: Simplify: 18

19 Day 5: Solving equations with Rational Exponents Steps to solve rational exponent equations: 1) Isolate the term with the rational exponent. 2) Raise it to its reciprocal power to undo the exponent. ) Isolate the variable and solve. 4) CHECK YOUR SOLUTIONS!! (If a solution does not work when you plug it back in, it is called extraneous) Solve the following equations: 1. 4x 2 5 = (7x ) 1 2 = 5. 2(x + 1) 2 = (2x + 4) 4 = x = 6. (x 2 + 5) =

20 7. x = x 7 6 = 2 9. (2x + 7) 1 2 = 10. (2x + 7) 1 2 x = x = (x 4) 2 4 = 5 20

21 Day 5 Homework: 21

Day 6: Radical Applications Radical Applications 1. Did you ever stand on a beach and wonder how far out into the ocean you could see? Or have you wondered how close a ship has to be to spot land?

22 Day 6: Radical Applications Radical Applications 1. Did you ever stand on a beach and wonder how far out into the ocean you could see? Or have you wondered how close a ship has to be to spot land? In either case, the function h d 2h can be used to estimate the distance to the horizon (in miles) from a given height (in feet). a. Cordelia stood on a cliff gazing out at the ocean. Her eyes were 100 ft above the ocean. She saw a ship on the horizon. Approximately how far was she from that ship? b. From a plane flying at 5,000 ft, how far away is the horizon? c. Given a distance, d, to the horizon, what altitude would allow you to see that far? 22

23 2. A weight suspended on the end of a string is a pendulum. The most common example of a pendulum (this side of Edgar Allen Poe) is the kind found in many clocks. The regular back-and-forth motion of the pendulum is periodic, and one such cycle of motion is called a period. The time, in seconds, that it takes for one period is given by the radical equation t 2 the force of gravity (10 m/s 2 ) and l is the length of the pendulum. l g in which g is a. Find the period (to the nearest hundredth of a second) if the pendulum is 0.9 m long. b. Find the period if the pendulum is m long. c. Solve the equation for length l. d. How long would the pendulum be if the period were exactly 1 s? 2

24 . When a car comes to a sudden stop, you can determine the skidding distance (in feet) for a given speed (in miles per hour) using the formula x in which s is skidding distance and x is speed. Calculate the speeding distance for the following speed. s 2 5x, a. 55 mph b. 65 mph c. 75 mph d. 40 mph e. Given the skidding distance s, what formula would allow you to calculate the speed in miles per hour? f. Use the formula obtained in (e) to determine the speed of a car in miles per hour if the skid marks were 5 ft long. 24

25 Solve each of the following applications. 4. The sum of an integer and its square root is 12. Find the integer. 5. The difference between an integer and its square root is 12. What is the integer? 6. The sum of an integer and twice its square root is 24. What is the integer? 7. The sum of an integer and times its square root is 40. Find the integer. 25

26 Day 7: ACT & Practice 26

27 Day 8: Study Guide I. Simplify Radicals x a x 10 y a 4 b 5 c x 5 y a b 7 II. Converting to and from radical form/rational exponents Write each expression in radical form. 1. (2y) 1 2. a 4. z m a (10n) 2 Write each expression in exponential form m 8. 2y 5 9. n 4 27

28 III. Solving Radical Equations 1. 2x + 1 = x + 2 = x + 6 = x 4 = 2. 4x 8 = x + 1 = x 1 = 8. 7x 6 5x + 2 = 0 28

29 IV. Solving Rational Equations 1. (x 2) 2 4 = 5 2. (7x ) 1 2 = 5. (2x + 4) 4 = x 2 5 = 10 29

30 V. Solving Radical Applications 8. The distance between the top of a lighthouse and s hip at sea can be found using the formula d h 2h where d is the distance to the horizon (in miles) and h is the height (in feet) of a given structure. d. The lighthouse at Cape Hatteras saw a ship on the horizon. Cape Hatteras Lighthouse is 19 feet tall. How fat away is the ship? e. The ship USS Awesome is on the horizon at a distance of 22 miles from a lighthouse off the coast of Cape Town, South Africa. How tall is the lighthouse in Cape Town? 9. The formula t 2 L 10 can measure the time it takes a wrecking ball to swing back and forth on a crane where t is the length of the period and L is the length of the wrecking ball. e. Find the period (to the nearest hundredth of a second) if the Wrecking ball has is 10 m long. f. How long would the wrecking ball be if the period were exactly 8 s?

Name: Period: Unit 3 Modeling with Radical and Rational Functions

Name: Period: Unit 3 Modeling with Radical and Rational Functions Name: Period: Unit Modeling with Radical and Rational Functions 1 Equivalent Forms of Exponential Expressions Before we begin today s lesson, how much do you remember about exponents? Use expanded form