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Lesson 2.1: Simplifying, Adding and Subtracting Radicals
Learning Goals:
- How do we simplify radical expressions?
- How do we add and subtract radical expressions?
Are the following numbers rational, irrational, or neither one?
- −√25 rational 2. √35 irrational
- √−8 neither 4. 3 √−8^ rational
Rational Numbers:
numbers that can be expressed in the form 𝑎𝑏 where 𝑏 ≠ 0
(i.e., fraction, integers, terminating decimals, repeating decimals)
Irrational Numbers:
numbers that cannot be expressed as a fraction.
index = 2 index = 3
√4 = 2^ √
√64 vs. √
(^3) to get the “3” use MATH #4 8 vs. 4
Directions: Express in simplest radical form.
- √32 2. 5√
index = 2 index = 2
√16 ∙ 2^ 5 ∙ √4 ∙ 10
4 ∙ √2 5 ∙ 2 ∙ √
10√
What is the difference between these two questions? Different index!
- √52𝑥^6 𝑏^5 index = 2 √52𝑥^36 𝑏^5 index = 3
√4 ∙ 13 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏 √2 ∙ 2 ∙ 13 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏^3
(groups of two) (groups of three)
2 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑏 ∙ 𝑏 ∙ √13 ∙ 𝑏 𝑥 ∙ 𝑥 ∙ 𝑏 ∙ √2 ∙ 2 ∙ 13 ∙ 𝑏 ∙ 𝑏^3
2𝑥^3 𝑏^2 √13𝑏 𝑥^2 𝑏 √52𝑏^32
- Based on the previous question, explain how you would solve the following problem? Then simplify the radical expression.
√32𝑎^43 𝑏^4
√2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏 ∙ 𝑏^4
2 ∙ 𝑏 ∙ √2 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎^4
2𝑏 √2𝑎^3
4
Important information to remember:
Radicals that are simplified have:
No fractions left under the radical symbol. No perfect power factors in the radicand, 𝑘. No exponents in the radicand, 𝑘, greater than the index, 𝑛. No radicals appearing in the denominator of a fractional answer.
Adding and Subtracting Radicals
- 4𝑥 − 𝑥 2. 3𝑥 + 2𝑥 − 9𝑥 3. 7𝑥 + 10𝑦 − 3𝑥
3𝑥 −4𝑥 10𝑦 + 4𝑥
What was your method for simplifying # 1 – 3?
Combine like terms! Same variable/exponent
How do you think it relates to our lesson today on adding and subtracting radicals?
Combine like radicals (terms) – Same index and same radicand.
To add or subtract radicals:
You must have the same index and radicand! Expand/Reduce radicals 1st.
Directions: Add or subtract the following, express all answers in simplest radical form.
- 5√11𝑥^3 − √11𝑥^3 2. 2√300 + 4√
4√11𝑥^3 cannot add until radicands are the same
2√100 ∙ 3 + 4√25 ∙ 3
2 ∙ 10 ∙ √3 + 4 ∙ 5 ∙ √
20√3 + 20√
40√
3. 4 √160𝑎^4 + √10𝑎^44 4. 𝑎√45 + √20𝑎^2 − 5√2𝑎
√2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 5 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎^4 + √2 ∙ 5 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎^4 𝑎√3 ∙ 3 ∙ 5 + √2 ∙ 2 ∙ 5 ∙ 𝑎 ∙ 𝑎 − 5√2𝑎
2𝑎 √2 ∙ 5^4 + 𝑎 √2 ∙ 5^4 3𝑎√5 + 2𝑎√5 − 5√2𝑎
2𝑎 √10^4 + 𝑎 √10^4 5𝑎√5 − 5√2𝑎
3𝑎 √10^4
1
9
4 − √45^ 6.^ √9𝑥
√ √4 +^
√ √4 − √9 ∙ 5^ √3 ∙ 3 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 + √3 ∙ 3 ∙ 3 ∙ 𝑥 ∙ 𝑥 1 2 +^
3 2 − 3√5^ 3𝑥√𝑥 + 3𝑥√ 4 2 − 3√
2 − 3√
Lesson 2.2: Multiplying and Dividing Radical Expressions
Learning Goals:
- How do we add and subtract radical expressions?
- How do we multiply and divide radical expressions?
Warm-Up:
Using your calculator, find the values of each of the following to two decimal places.
a. √3 ∙ √5 = 3.87 b. √15 = 3.
c. 2√2 ∙ 5√3 = 24.49 d. 10√6 = 24.
e. 4√6 ∙ 2√6 = 39.19 f. 8√36 = 39.
g. √10√2 = 2.24 h. √5 = 2.
i. 25√215√3 = 13.23 j. 5√7 = 13.
Based on the examples above, finish the rules below for multiplying/dividing radical expressions: when you multiply/divide, you do not need like radicals!
𝑛 √𝑎 (^) ∙ √𝑏 𝑛 (^) = 𝑛√𝑎 ∙ 𝑏 √𝑎 𝑛 𝑛 √𝑏 = √
𝑎 𝑏
𝑛
Multiplying/Dividing Racial Expressions
Express each of the following in simplest radical form:
3
4 3
3 Multiply 1st! 2.
(^3) √324𝑎 (^4) 𝑏 3 (^3) √6𝑎𝑏 (^2) Divide 1st!
4 3
3 √324𝑎
(^4) 𝑏 3 6𝑎𝑏^2
3
√28^3 (reduce?) √54𝑎^3 𝑏
3 (reduce?)
√7 ∙ 2 ∙ 2^3 (can’t)^ √3 ∙ 3 ∙ 3 ∙ 2 ∙ 𝑎 ∙ 𝑎 ∙ 𝑎 ∙ 𝑏^3
3𝑎 √2𝑏^3
4√2+8√ 2√2 separate into 2 fractions^ 4.^ 2𝑥√3𝑥
4√ 2√
8√ 2√
(2𝑥 ∙ 3)√(3𝑥^2 𝑦 ∙ 15𝑥𝑦^3 )
2 + 4√6 unlike terms! 6𝑥√3 ∙ 3 ∙ 5 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦 ∙ 𝑦
18𝑥^2 𝑦^2 √5𝑥
- (3√2 + √7)(3√2 − √7) = OR by a 2 𝑥 2 Table
9√4−3√14 + 3√14 − √49 =
9(2) − 7 =
18 − 7 =
11
- Express 2√6(8 − 4√3) as a product in simplest form.
16√6 − 8√18 = 16√6 − 8√9 ∙ 2 = 16√6 − 8(3)√2 = 16√6 − 24√
Lesson 2.3: Rationalizing a Denominator
Learning Goal: What do you think it means to rationalize the denominator of a fraction? Get rid of the radicals in the denominator!
Conjugate is a binomial formed by negating the second sign of the binomial
Example: 𝑥 − 3 and 𝑥 + 3 or 𝑥 + √5 and 𝑥 − √
(𝑥 − 3)(𝑥 + 3) = 𝑥^2 + 3𝑥 − 3𝑥 − 9 = 𝑥^2 − 9
(𝑥 + √5)(𝑥 − √5) = 𝑥^2 −𝑥√5 + 𝑥√5 − √25 = 𝑥^2 − 5
Rationalizing the denominator:
4
2√3 =^
4∙√
2√3∙√3 =^
4√
6 =^
2√ 3 2.^
3
𝑎^3 ∙
2∙2 =^
𝑎√2𝑎 2
Use the conjugate!
√
5−2√3 =^
√
5−2√3 ∙^
5+2√
5+2√3 =^
5√3+2√
25+10√9−10√9−4√9 =^
5√3+2(3)
25−4(3) =^
6+5√ 13
2+√
3+√2 =^
2+√
3+√2 ∙^
3−√
3−√2 =^
6−2√2+3√2−√
9−3√2+3√2−√4 =^
6+√2−
9−2 =^
4+√ 7
- Simplify:
3−√
3+√3 =^
3−√
3+√3 ∙^
3−√
3−√3 =^
9−6√3+
9−3 =^
12−6√
- The expression
3+4√ 4−2√5 is equivalent to
16 − 4(5)^
Homework 2.3: Rationalizing a Denominator
- The expression
3−√ √3 is equivalent to
(1)
√3−2√ √^
2
3 √6^ (3)^
3−√
3 (4)^ √3 −^
2
- Express
5 3−√2 with a rational denominator, in simplest radical form.
- Express the reciprocal of 3 − √7 in simplest radical form with a rational denominator.
- The expression
7 2+3√2 is equivalent to
(1)
−2+3√ 2 (2)^
2−3√
2 (3)^ −2 + 3√2^ (4)^ 2 − 3√
- Expressed in simplest form,
2√ 1−√
is equivalent to
(1) −3 − √3 (2) −3 + √3 (3) 2√3 (4) −
- The expression
3+5√ 4−2√3 is equivalent to
(1)
−9+7√ 2 (2)^
21+13√ 2 (3)^
−18+14√ 4 (4)^
42−26√ 4
A rational exponent does not have to be in the form
1 𝑛. Other rational numbers,
such as
3
2 or −^
4 5 can also be used as exponents.
Denominator = index
Let’s think about the expression 4
3 (^2). Evaluate this expression in two different
ways by using the laws of exponents. Then verify these answers with your calculator.
4
3 (^2) = (√4)
3 = 2^3 = 8
3 (^2) = √4^3 = √64 = 8
3 (^2) = 8
Rational Exponent Definition
For any rational number
𝑚
𝑛 we define^ 𝑏
𝑚 𝑛 (^) to be ( √𝑏^ 𝑛^ )𝑚 or 𝑛√𝑏^ 𝑚.
Rewrite each of the following using radicals, and then simplify, if possible.
3
4 = ( √16^4 )
3
= 2^3 = 8 2. 25
3
3
= 5^3 = 125
2
8
2 3
1 ( √8^3 )
2 =^
1 4
64 125
−^23
64 −
2 3 125 −
2 3
125
2 3 64
2 3
( √125^3 )
2
( √64^3 )
2 =^
25 16
5. (27𝑝^6 )
5
3 = ( √27𝑝^3 6 )
5
= (3𝑝^2 )^5 = 3^5 𝑝^10 = 243𝑝^10
Practice:
1. Yoshiko said that 16
1
4 = 4 because 4 is one-fourth of 16. Use properties of
exponents to explain why she is or is not correct.
She is wrong because 16
1
4 = √16^4 = 2 and 2 ≠ 4.
2. Rita said that 8
2
3 = 128 because 8
2
3 = 8^2 ∙ 8
1
3 , so 8
2
3 = 64 ∙ 2, and then
2
3 = 128. Use properties of exponents to explain why she is or is not correct.
She is wrong because 8
2 (^3) = (√8^3 )
2 = 2^2 = 4 and 4 ≠ 128.
3. Simplify the expression (𝑚^6 )−
2 (^3) and write your answer using a positive
exponent. Use the power law!
(𝑚^6 )−
2
2
3 = 𝑚−4^ =
𝑚^4
- Simplify the expression below and write your answer using positive exponents only.
1 (^2) 𝑦−
𝑦𝑥−
7 4
4
=
𝑥^2 𝑦−
𝑦^4 𝑥−^
𝑥^2 𝑥^7
𝑦^4 𝑦^8
𝑥^9
𝑦^12
5. Find the expression (𝑥 + 3)
1
2 + (𝑥 − 3)^0 + (𝑥 + 2)−
2 (^3) when 𝑥 = 6
1 (^2) + (6 − 3)^0 + (6 + 2)−
2 3
1 (^2) + 3^0 + 8−
2 3
2 3
(√8^3 )
or
or 4.
Lesson 2.5: Solving Radical Equations
Learning Goals
- How do we solve radical equations?
- How do we check for extraneous roots?
When solving a radical equation, the equation derived by squaring both sides is not equivalent to the given equation. In other words, the derived equation does not always have the same solution as the original equation. If a number is a root of the original equation, that number must also be a root of the derived equation, but the converse of this statement is not true.
It is helpful to demonstrate why an extraneous root occurs.
- (^) √𝑥 − 2 = 5
(√𝑥 − 2)
2 = 5^2 square both sides to eliminate the radical
𝑥 − 2 = 25 make sure to always check your answer:
𝑥 = 27 √27 − 2 = 5
√25 = 5
5 = 5√
√2𝑦 − 1 = −3^ isolate your radical
(√2𝑦 − 1)
2 = (−3)^2 make sure to always check your answer:
2𝑦 − 1 = 9 √2(5) − 1 + 7 = 4
2𝑦 = 10 (^) √9 + 7 = 4
𝑦 = 5 3 + 7 = 4
10 ≠ 4 therefore it is { } or ∅
3√𝑥 − 2 = 2√𝑥 + 8 make sure to always check your answer:
(3√𝑥 − 2)
2 = (2√𝑥 + 8)
2 3√10 − 2 − 2√10 + 8 = 0
9(𝑥 − 2) = 4(𝑥 + 8)^ 3√8 − 2√18 = 0
9𝑥 − 18 = 4𝑥 + 32 3√4 ∙ 2 − 2√9 ∙ 2 = 0
5𝑥 = 50 3 ∙ 2√2 − 2 ∙ 3√2 = 0
𝑥 = 10 6√2 − 6√2 = 0
0 = 0√
- 𝑥 = 1 + √𝑥 + 5 make sure to always check your answers:
𝑥 − 1 = √𝑥 + 5 isolate the radical 4 = 1 + √4 + 5
(𝑥 − 1)^2 = (√𝑥 + 5)
2 4 = 1 + √
(𝑥 − 1)(𝑥 − 1) = 𝑥 + 5 4 = 1 + 3
𝑥^2 − 1𝑥 − 1𝑥 + 1 = 𝑥 + 5 4 = 4√
𝑥^2 − 2𝑥 + 1 = 𝑥 + 5
𝑥^2 − 3𝑥 − 4 = 0 −1 = 1 + √−1 + 5
(𝑥 − 4)(𝑥 + 1) = 0 −1 = 1 + √
𝑥 − 4 = 0 and 𝑥 + 1 = 0 −1 = 1 + 2
𝑥 = 4 and 𝑥 = −1 −1 ≠ 3
