Download Radicals Lesson #1 - White Plains Public Schools

Chapter 3 Real Numbers and Radicals
Day 1: Roots and Radicals
A2TH SWBAT evaluate radicals of any index
Do Now: Simplify the following:
a) 22
b) (-2)2
c) -22
d) 82
e) (-8)2
f) 53
g)(-5)3
Based on parts a and b, what are the possibilities for the √4 ?
A square root of k is one of the two equal factors whose product is k. Every positive number has two square
roots.
Example: 1) x2 = 9
2) y2 = 5
The principal square root of a positive real number is its positive square root. In general, when referring to the
square root of a number, we mean the principal square root.
Example: √25 =
The cube root of k is one of three equal factors whose product is k. The cube root of k is written as
3
√𝑘 .
3
Since 4 ∙ 4 ∙ 4 = 64, 4 is one of the three equal factors whose product is 64 and √64 = 4.
There is only one real number that is the cube of 64. This real number is called the principal cube root.
Note: (−4) ∙ (−4) ∙ (−4) = −64. Therefore,
3
√−64 = −4 and – 4 is the principal cube root of –64.
The nth root of k is one of the n equal factors whose product is k.
The nth root of k is written as
k is the radicand
n is the index
𝑛
√𝑘 .
𝑛
√𝑘 is the radical
If k is positive, the principal nth root of k is the positive nth root
If k is negative and n is odd, the principal nth root of k is negative
If k is negative and n is even, there is no principal nth root of k in the set of real numbers
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When a variable appears in the radicand, the radical can be simplified if the exponent of the variable is
divisible by the index.
Example: Since 8x6 = (2x2)(2x2)(2x2),
3
√8𝑥 6 = 2x2.
Practice:
1) If a2 = 169, find all values of a .
2) Evaluate each of the following in the set of real numbers:
b. −√121𝑏 4
a. √49
c.
3
√−
1
d.
8
4
√81
3) Find the length of the longer leg of a right triangle if the measure of the shorter leg is 9 centimeters
and the measure of the hypotenuse is 41 centimeters.
4) State whether each of the following represents a number that is rational, irrational, or neither.
a. √25
b. √8
c. √−8
d.
3
√−8
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e. √0
f.
4
√16
g.
5
√−243
h. √0.25
Summary
The nth root of k is one of the n equal factors whose product is k.
The nth root of k is written as
k is the radicand
n is the index
𝑛
√𝑘 .
𝑛
√𝑘 is the radical
If k is positive, the principal nth root of k is the positive nth root
If k is negative and n is odd, the principal nth root of k is negative
If k is negative and n is even, there is no principal nth root of k in the set of real numbers
Exit Ticket: For what value(s) of x is the following radical a real number?
√𝟗 − 𝟑𝒙
Homework #1: Textbook page 87 #1, 2, 11-47 odd, 51
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Day 2: Simplifying Radicals
A2TH SWBAT simplify radicals
Do Now: Evaluate each of the following –
3
b) − √
a) √36
8
c) √144𝑥 2 𝑦10
27
d)
3
√−125𝑎6 𝑏18
How would you write √12 in simplest radical form?
In order to simplify a square root, we use the relationship √𝑎 ∙ 𝑏 = √𝑎 ∙ √𝑏
We usually want to use the greatest perfect square factor to write our answer in simplest radical form.
Simplest radical form is achieved when the radicand is a prime number or the product of prime numbers.
Example: √72 = √36 ∙ √2 = 6√2
Simplify: a) √24𝑥 4
b) √50𝑎3
c) √8𝑥 5 𝑦 6
d) 2√20𝑏17
We use the same process when simplifying a cube root, breaking down the radicand using a perfect cube
factor.
Example:
3
3
3
3
√24 = √8 ∙ √3 = 2 √3
Simplify: a)
3
√54
b)
3
√40𝑥 9
c)
3
√32𝑥 4
d)
3
√192𝑏14 𝑐12
Fractional Radicands: A radical is in simplest form when the radicand is an integer.
For any non-negative a and positive c: √
𝑎
𝑐
=
√𝑎
√𝑐
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Example: √
2
3
2
3
3
3
= √ ×
6
= √ =
9
√6
√9
=
b) √
4
√6
3
Fractional Radicands continued...
Simplify: a) √
d) √
9𝑎
8𝑏3
8
9
e)
4
√
1
2
5
c) √
f)
4
3
8
√
3𝑎
2𝑏3 𝑐 2
More Practice!
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Summary: For roots of any index:
𝑛
𝑛
𝑛
√𝑎𝑏 = √𝑎 ∙ √𝑏
Exit Ticket:
Simplify: a)
1
2
and
𝑛
𝑛
𝑎
√𝑎
√𝑏
√𝑏 =
√72𝑎𝑏 5
b)
𝑛
3
√375𝑥 5 𝑦 6
Homework #2: Textbook page 93 #1, 2, 3, 7, 9, 11, 17, 19, 21, 23, 27
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Day 3: Adding, Subtracting, and Multiplying Radicals
A2TH SWBAT add, subtract, and multiply radicals.
Do Now:
Simplify: a)
1
2
√72𝑎𝑏 5
b)
3
√375𝑥 5 𝑦 6
c) What are the rules when adding or subtracting algebraic terms? For example, how would you
simplify (b4-5b +3) - (3b4 + b + 3)?
d) What are the rules when multiply algebraic terms? For example, how would you simplify
(3a2b)(2abc)?
We use similar rules as above when adding, subtracting, and multiplying radical expressions.
To express the sum or difference of two radicals as a single radical, the radicals must have the same index and
the same radicand. In other words, they must be like radicals.
3
Example: 3 √4
3
3
+ 2 √4 = 5 √4
Simplify: a) 2√5 − 4√5
b) 4√3𝑏 + 3√3𝑏
c)
3
3
√2 + 7 √2
Two radicals that do not have the same radicand or do not have the same index are unlike radicals.
Example: √2 + √3
cannot be expressed as a single radical
Sometimes radical expressions that seem to be unlike terms can be simplified first and then appear as like
radicals.
Example: √8 + √50 = 2√2 + 5√2 = 7√2
Add or Subtract: a) √27𝑏 3 − √12𝑏 3
1
b) √12 + 9√3 + √8 − √72
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c)
3
3
3
√8𝑥 + √16𝑥 + √27𝑥
1
d) 3𝑥√
3𝑥
+ √300𝑥
e) Solve for x: 4𝑥 − √8 = √72
𝑛
Recall that if a and b are non-negative numbers, √𝑎𝑏 =
𝑛
𝑛
equality, √𝑎 ∙
√𝑏 =
𝑛
√𝑎 ∙
𝑛
√𝑏 . By the symmetric property of
𝑛
√𝑎𝑏. We can use this rule to multiply radicals. The Distributive Property may also
be used when needed.
Example: √8 ∙ √2 = √16 = 4
Multiply: a) √6𝑎3 ∙ √18𝑎
3
c) 5 √𝑥
∙
3
4 √8𝑥 7
b) 3√2 ∙ 4√10
4
d) √48𝑥 2
4
∙ √
𝑥2
3
e) √5 (3√6 + 12√3)
f) (3 + √3 ) (2 − √6 )
g) (2 + √2 )(5 − √2 )
h) (4 + √3 ) ( 4 − √3 )
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Summary
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Exit Ticket:
Simplify: (3 − √2)2
Homework #3: Textbook page 97 #1, 3, 7, 9, 21, 25, 29, 31, 33, 39; page 100 #21-45 odd
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Day 4: Dividing Radicals and Rationalizing a Denominator
A2TH SWBAT 1) simplify quotients of radical expressions and 2) simplify radical expressions by rationalizing
the denominator.
Do Now: Simplify the following radical expressions –
a)
3
3
3
3
−2 √56 + 5 √189 − 3 √7
b) ( √2
3
− 7)( √4 − 7)
Compare these two computations:
√9
√25
√
The general rule for the quotient of roots:
𝑛
√𝑎
𝑛
√𝑏
𝑛
𝑛
Simplify: a)
3
√
5
27
√4
√9
=
𝑛
If √𝑎 and √𝑏 are real numbers, with √𝑏 ≠ 0, then
𝑛
𝑎
= √
𝑏
This rule can also be used in the reversed order:
Example:
9
25
2
𝑛
𝑎
√𝑏 =
Example:
3
b) √
8𝑥 2
49
√8
√2
𝑛
√𝑎
𝑛
√𝑏
= √
8
2
= √4 = 2
c)
√75𝑥 3 𝑦
√3𝑥
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A fraction is in simplest form when the denominator is a rational number. We do not want a radical in the
denominator.
2
Example 1. Rationalize:
√7
2
√7
√7
√3𝑥
Simplify: a)
√2
b)
Example 2. Rationalize:
( 7) =
√
2√7
7
3
5√10
c)
√5𝑥 3
4
√−256
√80
d) 4
√5𝑥 4
3
√−2
3
2+ √6
Here we use what’s called the conjugate to rationalize the denominator.
The conjugate of (2 + √6) is (2 − √6).
3
2+ √6
Rationalize: a)
b)
c)
(
2− √6
6−3√6
2− √6
4−2√6+2√6− √36
)=
=
6−3√6
4−6
=
6−3√6
−2
= −
6−3√6
2
5
√11−7
6
√5+ √2
√3+3
5− √3
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d)
√𝑥+ √𝑦
√𝑥− √𝑦
e)
√𝑥+𝑦
3+ √𝑥+𝑦
f)
g)
6
√3
√5
√3𝑥
+
−
h) Solve for x:
8
√2
√3
√5𝑥
𝑥√2 = 3 − 𝑥
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Summary
Exit Ticket:
Homework #4 : Textbook page 103 #17-31 odd; page 107 #21-35 odd, 41-45 odd
Page 17
Day 5: Solving Radical Equations
A2TH SWBAT: solve radical equations
Do Now:
A radical equation is an equation in which the variable is hiding inside a radical sign.
To solve a radical equation, follow these steps:
1. Isolate the radical (or one of the radicals) to one side of the equal sign.
2. If the radical is a square root, square each side of the equation. (If the radical is not a
square root, raise each side to a power equal to the index of the root.)
3. Solve the resulting equation.
4. Check your answer(s) to avoid extraneous roots.
1.
x2 7  2
2. √2𝑥
−1 +5 = 2
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3.
4 + √1 − 3𝑥 = 12
5. √5𝑥
7. 𝑥
+ 3 = √3𝑥 + 7
− 3 = √30 − 2𝑥
4.
6. 𝑥
20  2 x  5x  8
− 1 = √5𝑥 − 9
8. 𝑥 = 1 + √15 − 7𝑥
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9. 2√𝑥
11.
+ 8 = 3√𝑥 − 2
2  3x  2  6
3
13. √𝑥
− 4 = √9𝑥
10. √𝑥
12.
+ 5 = √𝑥 2 − 15
5
√3𝑥 − 5 = −2
14. √𝑥 − √𝑥 − 5 = 1
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15.√2𝑥 + 3 = 5 − √𝑥 + 1
16. √2𝑥 + 1 − √𝑥 − 3 = 2
Summary
To solve radical equations:
1. Isolate the radical (or one of the radicals) to one side of the equal
sign.
2. If the radical is a square root, square each side of the equation.
(If the radical is not a square root, raise each side to a power equal
to the index of the root.)
3. Solve the resulting equation.
4. Check your answer(s) to avoid extraneous roots.
Exit Ticket:
Homework #5: page 112 #2, 11-35 odd
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A2TH
Packet #3:
Name:______________________________
Teacher:____________________________
Pd: _______
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Table of Contents
o
Day 1: SWBAT: evaluate radicals of any index
HW: Textbook page 87 #1, 2, 11-47 odd, 51
o
Day 2: SWBAT: simplify radicals
HW: Textbook page 93 #1, 2, 3, 7, 9, 11, 17, 19, 21, 23, 27
o
Day 3: SWBAT: add, subtract and multiply radicals
HW: Textbook page 97 #1,3,7,9,21,25,29,31,33,39; page 100 #21 – 45 odd
o
Day 4: SWBAT: 1) simplify quotients of radical expressions and 2) simplify radical expressions by
rationalizing the denominator.
HW: Textbook page 103 #17-31 odd; page 107 #21-35 odd, 41-45 odd
o
Day 5: solve radical equations
HW: page 112 #2, 11-35 odd
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Assignment #1
Assignment #2
Page 24
Assignment #3
Page 25
Assignment #4
Page 26
Assignment #5
Page 27