GEOMETRY
MODULE 2 LESSON 22
MULTIPLYING AND DIVIDING EXPRESSIONS WITH RADICALS
OPENING EXERCISE
Evaluate the following without a calculator.

42 = 16

102 = 100

√16 = 4

√100 = 10
RULES FOR SQUARE ROOTS
1. 𝑇ℎ𝑒 𝑖𝑛𝑑𝑒𝑥 (𝑟𝑜𝑜𝑡) 𝑓𝑜𝑟 √𝑛 𝑖𝑠 𝑖𝑚𝑝𝑙𝑖𝑒𝑑 𝑡𝑜 𝑏𝑒 2.
𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟, 𝑛, 𝑢𝑛𝑑𝑒𝑟 𝑡ℎ𝑒 𝑟𝑎𝑑𝑖𝑐𝑎𝑙 𝑠𝑦𝑚𝑏𝑜𝑙 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑡ℎ𝑒 𝑟𝑎𝑑𝑖𝑐𝑎𝑛𝑑.
2. √𝑥 = 𝑦 𝑖𝑓 𝑦 2 = 𝑥
3. √𝑎𝑏 = √𝑎 × √𝑏 𝑤ℎ𝑒𝑟𝑒 𝑎 𝑎𝑛𝑑 𝑏 𝑎𝑟𝑒 𝑛𝑜𝑛𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠
𝑎
√𝑎
4. √𝑏 =
𝑤ℎ𝑒𝑟𝑒 𝑎 𝑎𝑛𝑑 𝑏 𝑎𝑟𝑒 𝑛𝑜𝑛𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠
√𝑏
5. √−𝑛 𝑖𝑠 𝑎𝑛 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑛𝑢𝑚𝑏𝑒𝑟 𝑎𝑛𝑑 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑟𝑢𝑙𝑒𝑠 𝑤𝑖𝑙𝑙 𝑎𝑝𝑝𝑙𝑦.
6. 𝑇ℎ𝑒 𝑎𝑛𝑠𝑤𝑒𝑟 𝑡𝑜 𝑎 𝑠𝑞𝑢𝑎𝑟𝑒 𝑟𝑜𝑜𝑡 𝑚𝑎𝑦 𝑏𝑒 𝑏𝑜𝑡ℎ 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑎𝑛𝑑 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒.
𝑏
𝑎
7. √𝑥 𝑎 = 𝑥 𝑏
MOD2 L22
“The power is up above the roots are down below.”
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WORKBOOK EXERCISES
1. √172 = 17
2. √510 = √52 × √52 × √52 × √52 × √52 = 5 × 5 × 5 × 5 × 5 = 55
10
Optional: √510 = 5 2 = 55 Rule #7 applied
3. √4𝑥 4 = √4 × √𝑥 2 × √𝑥 2 = 2 × 𝑥 × 𝑥 = 2𝑥 2
4
Optional: √4𝑥 2 = √4 × 𝑥 2 = 2𝑥 2
4. Compare the value of √36 to the value of √9 × √4. What rule does this comply with?
√36 = 6
√9 × √4 = 3 × 2 = 6
Rule 3
Does the rule hold for 𝑎 = −4 and 𝑏 = −9?
√−4 × −9 = √36 = 6
√−4 × √−9 𝑦𝑖𝑒𝑙𝑑 𝑖𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦 𝑛𝑢𝑚𝑏𝑒𝑟𝑠!
Rule 3 does not hold.
100
5. Compare the value of √ 25 to the value of
√
√100
√25
. What rule does this comply with?
100
= √4 = 2
25
√100
√25
=
10
=2
5
Does the rule hold for 𝑎 = −100 and 𝑏 = −25?
No, the expression would yield imaginary numbers just like in exercise 4.
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A SQUARE ROOT VS THE SQUARE ROOT

THE SQUARE ROOT: The square root of a number always refers to the principle square root or
the positive root of the number.

A SQUARE ROOT: A square root of a number may be negative. Often used when solving a
quadratic equation. Solving √4 yields ± 2.

Example: −2 is a square root of 4, but the square root of 4 is 2.
RATIONALIZING THE DENOMINATOR
One reason we do this is so that we can better estimate the value of a number. We know that
√1 < √2 < √4 so the answer for √2 is between 1 and 2. √2 ≈ 1.414
What is the value of
1
√2
? Would it be easier to determine the value of
√2
2
? Well,
1
√2
=
√2
2
through
rationalizing the denominator.
To rationalize a denominator, multiply both numerator and denominator by a radical that will get rid of
the radical in the denominator. Hint: Bring the radicand to a power that is a multiple of two.
Practice
1
√3
1
√3
×
√3
√3
=
√3
√9
=
√3
3
Note: √9 = √32
1
√3
Can
√3
3
=
√3
3
be reduced? Why or why not?
No, the square root operation is yet to be taken in the numerator.
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SIMPLEST RADICAL FORM
An expression is in its simplest radical form when the radicand has no factor that can be raised to a
power greater than or equal to the index and there is no radical in the denominator.
To simplify a radical, find the largest factor of the radicand that is a perfect square.
Practice
√75
√25 × 3 = √25 × √3
5√3
ON YOUR OWN

Rationalize:
2
√10
2
∙
√10
√10 √10

=
2√10
√10
=
10
5
Simplify: √20
√20 = √4 ∙ 5 = √4 ∙ √5 = 2√5

Rationalize and simplify:
𝑥
√𝑥 3
𝑥
∙
√𝑥 3
√𝑥 3 √𝑥 3
=
𝑥 ∙ √𝑥 2 ∙ √𝑥 𝑥 ∙ 𝑥 ∙ √𝑥
√𝑥
=
=
3
3
𝑥
𝑥
𝑥
or
𝑥
√𝑥 3
∙
√𝑥 𝑥 ∙ √𝑥 𝑥 ∙ √𝑥
√𝑥
=
=
=
2
𝑥
𝑥
√𝑥
√𝑥 4
GROUP WORK
Complete as many of the exercises (6-17) in your workbook as possible.
HOMEWORK

Problem Set Module 2 Lesson 22 on page 168: #1-10, 13, and 14. Extra Credit #11

Due: Tuesday, Dec 6 (Corrections)
Thursday, Dec 8 (No Corrections)

All work must be present on a separate sheet of paper stapled to the original homework.
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