M2
Lesson 22B
NYS COMMON CORE MATHEMATICS CURRICULUM
Name____________________________
Date___________________Period_________
Lesson 22B: Rationalize the Denominator
Learning Targets
1. I can multiply and divide expressions that contain radicals to simplify their answers.
2. I can rationalize the denominator of a number expressed as a fraction.
Opening Exercises
1) Use your calculator and/or what you remember about radicals to simplify each of the following.
a) √6 ⋅ √60 =
b) √180 =
Rule 4:
Rule 4:
a  a
n
an  a
n 1
n
2
c) 3 √74 =
for nonnegative numbers 𝑎 , and even numbers 𝑛
 a a
n 1
2
a
for nonnegative numbers 𝑎 , and odd numbers 𝑛
Use the rule above to simplify the following radicals: (assume all variables are nonnegative)
a)
2610
b)
x 30
c)
217
d)
25x 21
Rationalizing the Denominator
Rule 5:
b
b
a b a



a
a
a a
for nonnegative numbers a and real numbers b
Use the rule above to simplify the denominators of each of the following:
a)
6
5
b)
15
5
c)
5
3
d)
1
3
Lesson 22B
NYS COMMON CORE MATHEMATICS CURRICULUM
Name____________________________
Date___________________Period_________
Why do we rationalize denominators?
 Makes estimating the value of a number easier:
If we know that √2 ≈ 1.414, then what is the value of

M2
By rationalizing the denominator we can see that
1
√2
1
√2
?
is equivalent to ____________________________
Which value is easier to estimate?
Reduce each radical expression to simplest radical form using the appropriate rules: (𝑥 ≥ 0)
1) √
4
𝑥4
𝟒. √
=
𝟏𝟕
𝟐𝟓
3) √
2) √32𝑥 =
=
5)
√𝟑𝟔
√𝟏𝟖
=
1
3
=
6. √
𝒙𝟓
𝟐
=
7. Nazem and Joffrey are arguing about who got the right answer. Nazem says the answer is
answer is
√𝟑
.
𝟑
𝟏
, and Joffrey says the
√𝟑
Show and explain that their answers are equivalent.
Lesson summary
Rationalizing the Denominator simply means to remove all radicals from the denominator of a fraction without
changing the value of the fraction.
Lesson 22B
NYS COMMON CORE MATHEMATICS CURRICULUM
Name____________________________
M2
Date___________________Period_________
Lesson 22B: Rationalize the Denominator
Exit Ticket/ Class work
Write each expression in its simplest radical form.
1. √243
2
4. √ =
7
7
√27
√3𝑥
=
2. √ =
3.
5. √98𝑥 2 =
6. √85𝑥10 𝑦 5 =
5
7. Teja missed class today. Explain to her how to write the length of the hypotenuse in simplest radical form.
8. Demarcus found the scale factor of a dilation to be
was
1
. When he compared his answer to Yesenia’s, which
√2
√2
, he told her that one of them must have made a mistake.
2
to Demarcus and Yesenia that proves they are both correct.
Show work and provide an explanation
Lesson 22B
NYS COMMON CORE MATHEMATICS CURRICULUM
Name____________________________
M2
Date___________________Period_________
9. Is Woodstock correct in the comic pictured below? Explain his steps in your own words.
___________________________________________________________________________________
___________________________________________________________________________________
___________________________________________________________________________________
___________________________________________________________________________________
10. The captain of a ship recorded the ship’s coordinates, then sailed north and then west, and then recorded
the new coordinates. The coordinates were used to calculate the distance they traveled, √𝟓𝟕𝟖 km. When
the captain asked how far they traveled, the navigator said, “About 𝟐𝟒 km.” Is the navigator correct?
Under what conditions might he need to be more precise in his answer?
11. Simplify each expression by rationalizing the denominator
a.
√2
√3
b.
1
c.
2√2
12. Simplify each expression
a. √50𝑟 2 𝑠 4
b. 3√18𝑥 3
c. √8𝑎3 𝑏 5
d. √50𝑥 3 𝑦 2
7
√11
Lesson 22B
NYS COMMON CORE MATHEMATICS CURRICULUM
Name____________________________
M2
Date___________________Period_________
Lesson 22B: Rationalize the Denominator
Homework
1. Sarahi found the distance between two points on a coordinate plane to be √𝟕𝟒. Can this answer be
simplified? Explain.
2. Pablo found the length of the hypotenuse of a right triangle to be √𝟒𝟓. Can the length be simplified?
Explain.
3. Determine the area of a square with side length 𝟐√𝟕 in.
1
4. Calculate the area of the triangle to the right. Remember, 𝐴 = 2 𝑏ℎ
5. √16𝑥 3 =
7.
√
63
5x
=
6.
√
90
𝑏6
=
8. √(−15)2 𝑥 8 =