NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 9
M1
ALGEBRA II
Lesson 9: Radicals and Conjugates
Classwork
Opening Exercise
Which of these statements are true for all 𝑎, 𝑏 > 0? Explain your conjecture.
i.
ii.
iii.
2(𝑎 + 𝑏) = 2𝑎 + 2𝑏
𝑎+𝑏
2
=
𝑎
2
+
𝑏
2
√𝑎 + 𝑏 = √𝑎 + √𝑏
Example 1
Express √50 − √18 + √8 in simplest radical form and combine like terms.
Exercises 1–5
1
9
1.
√
2.
√2 (√3 − √2)
4
+ √ − √45
4
Lesson 9:
Radicals and Conjugates
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This file derived from ALG II-M1-TE-1.3.0-07.2015
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Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA II
3.
4.
5.
√
3
3
8
√
5
32
3
√16𝑥 5
Example 2
Multiply and combine like terms. Then explain what you notice about the two different results.
(√3 + √2) (√3 + √2)
(√3 + √2) (√3 − √2)
Lesson 9:
Radicals and Conjugates
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
S.46
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 9
M1
ALGEBRA II
Exercise 6
6.
Find the product of the conjugate radicals.
(√5 + √3)(√5 − √3)
(7 + √2)(7 − √2)
(√5 + 2)(√5 − 2)
Example 3
Write
√3
5−2√3
in simplest radical form.
Lesson 9:
Radicals and Conjugates
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
S.47
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA II
Lesson Summary

For real numbers 𝑎 ≥ 0 and 𝑏 ≥ 0, where 𝑏 ≠ 0 when 𝑏 is a denominator,
√𝑎𝑏 = √𝑎 ⋅ √𝑏 and

√𝑎
√𝑏
=
√𝑎
√𝑏
.
For real numbers 𝑎 ≥ 0 and 𝑏 ≥ 0, where 𝑏 ≠ 0 when 𝑏 is a denominator,
3
3
3
√𝑎𝑏 = √𝑎 ⋅ √𝑏 and

3 𝑎
√𝑏
3
=
√𝑎
3
√𝑏
.
Two binomials of the form √𝑎 + √𝑏 and √𝑎 − √𝑏 are called conjugate radicals:
√𝑎 + √𝑏 is the conjugate of √𝑎 − √𝑏, and
√𝑎 − √𝑏 is the conjugate of √𝑎 + √𝑏.
For example, the conjugate of 2 − √3 is 2 + √3.

To rewrite an expression with a denominator of the form √𝑎 + √𝑏 in simplest radical form, multiply the
numerator and denominator by the conjugate √𝑎 − √𝑏 and combine like terms.
Problem Set
1.
2.
Express each of the following as a rational number or in simplest radical form. Assume that the symbols 𝑎, 𝑏, and 𝑥
represent positive numbers.
a.
√36
b.
√72
c.
√18
d.
√9𝑥 3
e.
√27𝑥 2
f.
3
g.
3
h.
√9𝑎2 + 9𝑏 2
√16
√24𝑎
Express each of the following in simplest radical form, combining terms where possible.
a.
√25 + √45 − √20
b.
3√3 − √ + √
c.
3
d.
3
4
1
3
3 1
3
√54 − √8 + 7 √4
3
5
8
3
8
9
√ + 3√40 − √
Lesson 9:
Radicals and Conjugates
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
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This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
M1
ALGEBRA II
3.
Evaluate √𝑥 2 − 𝑦 2 when 𝑥 = 33 and 𝑦 = 15.
4.
Evaluate √𝑥 2 + 𝑦 2 when 𝑥 = 20 and 𝑦 = 10.
5.
Express each of the following as a rational expression or in simplest radical form. Assume that the symbols 𝑥 and 𝑦
represent positive numbers.
6.
a.
√3(√7 − √3)
b.
(3 + √2)
c.
(2 + √3)(2 − √3)
d.
(2 + 2√5)(2 − 2√5)
e.
(√7 − 3)(√7 + 3)
f.
(3√2 + √7)(3√2 − √7)
g.
(𝑥 − √3)(𝑥 + √3)
h.
(2𝑥√2 + 𝑦)(2𝑥√2 − 𝑦)
2
Simplify each of the following quotients as far as possible.
a.
(√21 − √3) ÷ √3
b.
(√5 + 4) ÷ (√5 + 1)
c.
(3 − √2) ÷ (3√2 − 5)
d.
(2√5 − √3) ÷ (3√5 − 4√2)
1
𝑥
7.
If 𝑥 = 2 + √3, show that 𝑥 + has a rational value.
8.
Evaluate 5𝑥 2 − 10𝑥 when the value of 𝑥 is
9.
Write the factors of 𝑎4 − 𝑏 4 . Express (√3 + √2) − (√3 − √2) in a simpler form.
2−√5
2
.
4
4
10. The converse of the Pythagorean theorem is also a theorem: If the square of one side of a triangle is equal to the
sum of the squares of the other two sides, then the triangle is a right triangle.
Use the converse of the Pythagorean theorem to show that for 𝐴, 𝐵, 𝐶 > 0, if 𝐴 + 𝐵 = 𝐶, then √𝐴 + √𝐵 > √𝐶,
so that √𝐴 + √𝐵 > √𝐴 + 𝐵.
Lesson 9:
Radicals and Conjugates
This work is derived from Eureka Math ™ and licensed by Great Minds. ©2015 Great Minds. eureka-math.org
This file derived from ALG II-M1-TE-1.3.0-07.2015
S.49
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.