Simplifying Radicals:Part I

To simplify a radical in which the radicand contains a perfect square as a factor

Example: √729 √9 ∙√81 3 ∙ 9Perfect Square! 27

Vocabulary and Key Concepts

xRadical symbol

Radicand

Read “the square root of x.”

NOTE: The index 2 is usually omitted when writing square roots.

2

Index

Table of Perfect Squares

12 = _____ 62 = _____ 112 = _____ 162 = _____

22 = _____ 72 = _____ 122 = _____ 172 = _____

32 = _____ 82 = _____ 132 = _____ 182 = _____

42 = _____ 92 = _____ 142 = _____ 192 = _____

52 = _____ 102 = _____ 152 = _____ 202 = _____

1 36 121 256

4 49 144 289

9 64 169 324

16 81 196 361

25 100 225 400

Complete the table below:

You may find the following table of perfect squares to be helpful when you are required to simplify square roots.

MENTAL MATH: Find two factors of 72, one of which is the greatest perfect square factor. Establish order, so that you don’t omit any!

1, 72

2,36

3, 24

4, 18

6, 12

8, 9

9,8 (once you have a repeated factor pair, you know that you have found ALL factors!)

Simplifying Square Roots

72g36 2

6 2

ALERT! Check to be sure you have simplified completely:

Simplifying Square Roots: An Alternate Method

72g8 9

g g4 2 9

2 3 26 2

NOTE: If you have a perfect square

(or perfect square factor) remaining under the

radical symbol, you have not simplified

completely.

Simplifying Square Roots

KEY: L K for perfect squares or perfect

square factors.

20 18

27 32

NOTE: If you have a perfect square (or perfect square factor) remaining under the radical symbol, you have not simplified completely.

Simplifying Radicals:Part II

To multiply, then simplify squareroots when possible

Product of Square Roots

PRODUCT OF SQUARE ROOTS For all real numbers x ≥ 0, y ≥ 0,

√x ∙√x = √x2 = x √x ∙√y = √x∙yNOTE: Squaring a number and finding

the square root are inverse operations.

Multiplying Square Roots with Common Radicands

√3 ∙√3 = (√3)2 = __

√4 ∙√4 = ____ = __

√5 ∙√5 = ____ = __

(2√3)2 = _ ∙ _ = __

(3√5)2 = _ ∙ _ = __

(2√5)2 = _ ∙ _ = __

3

(√4)2 4

(√5)2 5

4 3 12

9 5 45

4 5 20

NOTE: Squaring a number and finding the square

root are inverse operations.

Multiplying Square Roots with Different Radicands

√3 ∙√6 = √18 = ____ ________

√2 ∙√10 = ____ = ____________

√4 ∙√20 = ____ = ____________

(2√3) (5√3) = ____ = ____________

(3√2) (5√2) = ____ = ____________

(3√2) (2√6) = ____ = ____________

(5√3) (√6) = ____ = ____________

√9∙2=

3√2

√20 √4∙5 =

4√5√80 √16∙5 =

2√5

10∙3 30

6∙2 12

6 ∙ √12 5 ∙ √18

Dividing Radicals

To simplify an expression

containing a quotient of radicals

Quotient of Square Roots

QUOTIENT OF SQUARE ROOTS

For all real numbers x ≥ 0, y > 0

x x= yy

Simplify each expression:

18

6

a.

b. 243

Radical in

Denominator

Fraction

Under √

Rationalizing Denominators:1

2

1

2 2

2g

2

2

Rationalize

1

ALERT! When you rationalize, you are changing

the form of the number, but not its value.

Double Check:

1. Fraction under √ ?

2. Radical in

Denominator?

More Quotients of Radicals3

4

3

6

1

12

9

7

5

10

Summary

A radical expression is in simplest form when

each radicand contains no factor, other than one, that is a perfect square

the denominator contains no radicals and

each radicand contains no fractions.

Final Checks for Understanding

1. Simplify: √3 ∙√12

2. Simplify: √2 ∙√32

3. Indicate why each expressions is not in simplest radical form.

a.) 5x2 b.)√8y c.) √3x 5y 7

25

Homework Assignments:

DAY 1: Simplifying Radicals WS

DAY 2: Multiplying and Dividing Radicals WS