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1
Math 2
UNIT 3:
Radicals
2
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
3
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 _______.
4
Practice simplify the following radicals:
1. √12 2. √243
3. √48𝑥64
4. √16𝑥2 5. √80𝑛53 6. √96𝑥94
4. √−403
5. √18x43 6. √−32𝑥3𝑦65
5
Day 1 Homework:
Simplify the following.
1. √10003
2. √−1623
3. √128n84
4. √224r75 5. √−16a3b83
6. √448𝑥7𝑦96
7. √405x3y24 8. √512𝑥3 9. √56x5y
3
6
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
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. 3. Write the steps to solving the equation to the side of your cut outs.
8
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
9
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
10
7. √2x + 13
= √83
8. √3𝑥 − 2 = −5
9. √7𝑥 − 6 − √5𝑥 + 2 = 0 10. √5𝑥 − 23
= 8
11
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
12
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
13
10. √𝑥 − 63
= −5 11. √2𝑥 + 64
+ 7 = 11 12. 2√𝑥 + 75
− 15 = −9
13. √5𝑥 + 43
= 4
14. √216𝑥53
15. √375𝑥9𝑦73
14
Day 3 Homework:
15
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 _________________.
16
Examples:
1. (𝟑𝒙𝟓𝒚𝟖)𝟐 2. (−𝟔𝒙𝟖𝒚−𝟑)𝟐 3. ((−𝟑𝒙−𝟓)
(𝟏𝟐𝒙−𝟑))𝟐
Practice:
Convert from Radicals to Rational Exponents:
17
18
Day 4 Homework:
Simplify:
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.
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
20
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
21
Day 5 Homework:
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 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?
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 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?
24
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.
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 3 times its square root is 40. Find the integer.
26
Day 7: ACT & Practice
27
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
28
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
29
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?
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