Simplifying Radical Expressions: Product Property and Quotient Rule, Study notes of Algebra

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Notes 10-1A Simplifying Radical

Expressions

I. Product Property of Square Roots

80

50

125

450

16 * 5

25 * 2

25 * 5

225 * 2

4 5

5 5

Perfect Square Factor * Other Factor

LEAVE IN RADICAL FORM

Ex 3:

Ex 4:

Ex 5:

Ex 6:

Method 2: Pair Method

Sometimes it is difficult to recognize perfect squares within a number. You will get better at it with more practice, but until then, here is a second method:

-Break the radicand up into prime factors -group pairs of the same number -simplify -multiply any numbers in front of the radical; multiply any numbers inside of the radical

Example 2:

2 ∗ 2 ∗ 2 ∗ 2 ∗ 3 Step 1: Break up into primefactors Step 2: Group together any pairs

Step 4: Multiply numbers in front of radical; multiply numbers inside radical

2 ∗ 2 ∗ 2 ∗ 2 ∗ 3

Simplify 48

2 ∗ 2 3

22 22 3 Step 3: Simplify

 4 3

Ex 2:

40 

More Examples:

4 10  2 10

7 75  7 25 3  7 5 3^ ^35

4 10 

7 25 3 

Ex 3:

Ex 4:

B. Using Product Property to Multiply Square Roots Ex 1: Multiply 3 ∗ 15

3 ∗ 3 ∗ 5 32 ∗ 5

3 5

Method 2: Multiply together first 3 ∗ 15 45 9 ∗ 5 3 5

3 ∗ 15

Method 1: Break down and simplify

Practice

5 ∗ 10 2 ∗ 3 ∗ 6 ∗ 8 50 6 48 2 ∗ 25 6 16 ∗ 3 5 2 6 ∗ 4 3

Example 2: Mulitply 5 ∗ 10 Example 3: Multiply 2 6 ∗ 3 8

24 3

More Examples:

1. 30 a

34  a

34  30

 a

17 30

2. 54 x

4 y

5 z

(^7)  9 x^4 y^4 z^6  6 yz

 3 x

2 y

2 z

3 6 yz

A. Examples

Remember!!!!!

16 81



Examples:

2 5

4 9

45 49



If and are real numbers and 0, then a b

a b b a b

 

III. Quotient Rule for Square Roots

2 25



9 5 7

 (^)  3 5 7

16 81

^2 25



45 49



Ex 1: Ex 2:

Ex 3:

15 3



90 2



3 5 3

 (^)  3 5 3

 (^)  5

9 10 2

 (^)  9 2 5 2

  (^)  9 2 5 2

  (^)  3 5

Ex 4:

Ex 5:

Ex 3: Simplify (^483)

48 3 16 ∗ 3 3 4 3 3 4

Ex 4: Simplify

7 7 25

y 

8

x

7 6 25

 y y 

(^3 )

5

y y

8

x

^3

8 x

x

Ex 4:

Ex 5: