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Math 2

UNIT 3:

Radicals

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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.

Date Lesson Quiz Topic Homework

Monday 2/20

Simplifying Radicals

Tuesday 2/21 Solving Radical Equations

Wednesday 2/22 Solving Radical Equations

Thursday 2/23 Exponential Rules Quiz on Days 1 – 2

Friday 2/24 Solving with Rational

Exponents

Monday 2/27

Radical Applications

Tuesday 2/28

ACT Practice Day

Wednesday 3/1 Review

Thursday 3/2 Unit Test

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Day 1: Simplifying Radicals

Radicals:

√𝒙𝒏

= 𝒓

√643

= 4 (because 4*4*4=64)

index=___

radicand=____

root=____

Rewrite the expression using radical notations:

𝑥𝑓

𝑛 = √𝑥𝑛 𝑓

___ is the index and ___ is the exponent (power).

Examples:

1. 95/3 _________ 2. 4y1/5___________ 3. (2x)2/3 ____________

You Try:

4. 33/4 ________ 5. X1/7 __________ 6. (3y)4/5 ____________

Simplifying Radical Expressions:

Examples:

1. √243𝑦55 2. √1296𝑚4𝑛84

3. √144𝑣8

***If there is no index, it is understood

to be _______.

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Practice simplify the following radicals:

1. √12 2. √243

3. √48𝑥64

4. √16𝑥2 5. √80𝑛53 6. √96𝑥94

4. √−403

5. √18x43 6. √−32𝑥3𝑦65

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Day 1 Homework:

Simplify the following.

1. √10003

2. √−1623

3. √128n84

4. √224r75 5. √−16a3b83

6. √448𝑥7𝑦96

7. √405x3y24 8. √512𝑥3 9. √56x5y

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Day 2: Solving Radical Equations

Warm-Up

Simplify.

1. √120𝑥7 2. Write in radical form x3/2

3. 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).

3) Isolate the variable and solve.

4) Check for Extraneous solutions

Examples:

1. √4𝑥 − 8 = 0 2. √𝑥 + 6 = 𝑥

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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. 3. Write the steps to solving the equation to the side of your cut outs.

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Practice solving radical equations:

1. 5√𝑥 + 2 = 12 2. √2x + 13

= √83

3. √2x − 43

= −2 4. √12𝑥 + 13 = 2𝑥 + 1

5. √𝑥 − 2 − √𝑥 = 1 6. √3 − x5

+ 4 = 3

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Day 2 Homework:

Solve for x

1. √𝑥 − 3 − √𝑥 = 3 2. √3𝑥 − 2 = −5

3. √x − 23

= 4 4. √2𝑥 − 5 = 9

5. √𝑎 + 2 − 2 = 12 6. √3𝑥 + 14

− 5 = 0

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7. √2x + 13

= √83

8. √3𝑥 − 2 = −5

9. √7𝑥 − 6 − √5𝑥 + 2 = 0 10. √5𝑥 − 23

= 8

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Day 3: Solving Radical Equation cont.

Warm Up

Simplify the following:

1. −12√6250𝑥54 2. 3𝑥2𝑦 √2058𝑥3𝑦23

Solve the equations:

3. √𝑥 − 3 − 9 = 3 4. √3𝑥 + 14

− 5 = 0

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Day 3: Simplifying and Solving Radicals Practice

1. √80 2. √180 3. √1500

4. √3𝑥 + 6 = 9 5. √5𝑥 + 7 = 18 6. 3√𝑥 − 5 − 4 = 14

7. √128𝑥43 8. √56𝑥6𝑦84

9. √160𝑥8𝑦3𝑧55

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10. √𝑥 − 63

= −5 11. √2𝑥 + 64

+ 7 = 11 12. 2√𝑥 + 75

− 15 = −9

13. √5𝑥 + 43

= 4

14. √216𝑥53

15. √375𝑥9𝑦73

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Day 3 Homework:

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Day 4: Exponents

Converting from Radicals to Rational Exponents:

√𝑥𝑛 𝑓

= 𝑥𝑓

𝑛

Examples:

1. √𝑡2 3

= ______ 2. √5𝑥3 5

= ______ You Try. √2𝑥9

= _____

Review of Exponent Rules:

Multiplying Monomials with like bases

𝟑𝟐 ∙ 𝟑𝟒 = _______ 𝒚𝟒 ∙ 𝒚𝟏𝟎 = _______ 𝒙𝟑 ∙ 𝒙𝟏𝟒 = _______

When multiplying monomials with like bases, we _________ the exponents.

Raising a power to a power

(32)6=_____ (x3)5=_____ (2x4)3=_____

When raising a power to a power, we _______________ the exponents.

Dividing monomials with like bases

𝟑𝟔

𝟑𝟐 = ______ 𝒙𝟖

𝒙𝟑 = _______ 𝒚𝟓

𝒚𝟏𝟐 = ________

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 _________________.

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Examples:

1. (𝟑𝒙𝟓𝒚𝟖)𝟐 2. (−𝟔𝒙𝟖𝒚−𝟑)𝟐 3. ((−𝟑𝒙−𝟓)

(𝟏𝟐𝒙−𝟑))𝟐

Practice:

Convert from Radicals to Rational Exponents:

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Day 4 Homework:

Simplify:

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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.

3) 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. 4𝑥3

2 − 5 = 103 2. (7𝑥 − 3)1

2 = 5

3. 2(𝑥 + 1)3

2 = 54 4. 3(2𝑥 + 4)4

3 = 48

5. 𝑥1

4 − 2 = 3 6. 3 (𝑥2

3 + 5) = 207

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7. 𝑥1

2 − 5 = 0 8. 4𝑥7 − 6 = −2

9. (2𝑥 + 7)1

2 = 3 10. (2𝑥 + 7)1

2 − 𝑥 = 2

11. 3𝑥4

3 + 5 = 53 12. (𝑥 − 4)2

3 − 4 = 5

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Day 5 Homework:

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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 hhd 2 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 35,000 ft, how far away is the horizon?

c. Given a distance, d, to the horizon, what altitude would allow you to see

that far?

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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 g

lt 2 in which g is

the force of gravity (10 m/s2) and l is the length of the pendulum.

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 0.049 m long.

c. Solve the equation for length l.

d. How long would the pendulum be if the period were exactly 1 s?

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3. 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 xxs 52 ,

in which s is skidding distance and x is speed. Calculate the speeding

distance for the following speed.

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 35 ft long.

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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 3 times its square root is 40. Find the integer.

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Day 7: ACT & Practice

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Day 8: Study Guide

I. Simplify Radicals

1. √169𝑥4 2.√125𝑎63 3.√32𝑥10𝑦155

4. √48𝑎4𝑏5𝑐7 5. √243𝑥5𝑦155 6. √54𝑎3𝑏73

II. Converting to and from radical form/rational exponents

Write each expression in radical form.

1. (2𝑦)1

3 2. 3𝑎3 4⁄ 3. 𝑧2

3

4. 5𝑚2

5⁄ 5. 𝑎1.6 6. (10𝑛)3

2

Write each expression in exponential form.

7. √𝑚3

8. √2𝑦3 2 9. 3√𝑛45

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III. Solving Radical Equations

1. √2𝑥 + 13

= √83

2. √𝑥 + 6 = 13

3. √4𝑥3

− 8 = 0

4. 4√𝑥 − 34

− 13 = 3

5. 5√𝑥 + 2 = 12

6. √2𝑥 − 43

= −2

7. √12𝑥 + 13 = 19

8. √7𝑥 − 6 − √5𝑥 + 2 = 0

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IV. Solving Rational Equations

1. (𝑥 − 2)2

3 − 4 = 5

2. (7𝑥 − 3)1

2 = 5

3. 3(2𝑥 + 4)4

3 = 48

4. 4𝑥3

2 − 5 = 103

V. Solving Radical Applications

8. The distance between the top of a lighthouse and s hip at sea can be found using the formula hhd 2

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 193 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 10

2L

t 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?