Objective

Students will add, subtract, multiply, divide, and simplify radicals.

Warm-up

• Define what a perfect square is using your own words.

• List all of the perfect squares that you can remember off the top of your head.

Simplifying Radicals

• Simplifying radicals is like creating factor trees, but we are looking for perfect square factors.

• Example:

€

8 Are there any perfect squares that are factors of 8? What are they?

€

€

4

€

2

€

€

2 2

x

Let’s try these together:

€

A. 63

€

B. 24

€

C. 306

€

D. 1024

Answers to Simplifying Radicals - Question A

€

€

A. 3 7

Answers to Simplifying Radicals Question B

€

24

€ €

€

6€

€

4€

€

2 6

Answers to Simplifying Radicals Question C

€

€

306

€

€

9€

€

34

€

3 34

3+0+6 = 9, 306 is divisible by 9 too!

Answers to Simplifying Radicals Question D

€

1024

32

€

16

€

64

€

4 • 8

Adding and Subtracting with Radicals

• Adding and subtracting with radicals is a lot like combining like terms. You can only add or subtract “like” radicals.

• Example: €

€

3 2 + 4 2 + 5 3

“Like” radicals: Add the coefficients!

€

7 2 + 5 3 Final answer!

Try these on your own!

€ €

A. 3 2 − 5 2 − 3 2 + 2

€

B. 3 − 2 3 + 4 5 − 3 5€

€

C. 20 + 5 5 + 5 2

Question A Add/Subtract

€

A. 3 2 − 5 2 − 3 2 + 2

Since each term contains radical 2, just

combine the coefficients!

€ €

€

−4 2

Question B Add/Subtract

€

B. 3 − 2 3 + 4 5 − 3 5

“like” radicals

“like” radicals

€

€

− 3 + 5

Question C Add/Subtract

€

C. 20 + 5 5 + 5 2

€

€

2 5 + 5 5 + 5 2 Simplify radicals that you can simplify.

€

€

7 5 + 5 2 Combine “like” radicals.

Multiplying with Radicals• When multiplying a radical by a

radical:– multiply the coefficients– multiply the radicals– then simplify the radical if possible.

• Example: €

€

4 5 3 2( )

€

4( ) 3( ) 5( ) 2( )

€

12 10 Final Answer?

What property?

Try these :

€ €

A. 4 3 2 12( )

€ €

B. 5 25 3 100( )

€

€

C. 3 2( ) 4 − 2 3( )

€

€

D. 5 2( )2

Question A Multiplication

€

A. 4 3 2 12( )

€

8 36

8 • 6

48

Multiply

Simplify

Final Answer

Question B Multiplication

€

B. 5 25 3 100( )

€

B. 5 25 3 100( )

Method 1 Method 2

€

5• 3• 25 • 100

15 • 2500

15 • 50

750

€

5 • 5 3• 10( )

25 30( )

750

Question C Multiplication

€

C. 3 2( ) 4 − 2 3( )

€

€

3 2 • 4( ) − 3 2 • 2 3( )

12 2 − 6 6

Distributive PropertyMultiply and Simplify

Question D Multiplication

€

D. 5 2( )2

Which answer did you choose?

€

€

Choice A - 25 4

€

€

Choice B - 25 2

€

Choice C - 50€

€

5 2( ) 5 2( )

25 4

25 2( ) = 50

Dividing with Radicals

€

€

64

16=

64

16= 4 = 2

64

16=

8

4= 2

€

€

50

10=

25 2

5 2=

25

5=

25

5= 5

50

10=

50

10= 5

Let’s kick it up a notch…Simplify using radicals.

€ €

A. 50

5

€ €

B. 63

54

€

€

C. 32

36

€

€

D. 5 + 50

5

Question A - Division

€

A. 50

5

€

€

25 5

5€

€

5 5

5

€

€

5

Simplify the radical.Take the square roots of perfect squares.

Divide by 5. This is your final answer!

Question B - Division

€

B. 63

54

Choice #1 = 1.08

Choice #2 = €

€

7

6

Choice #3 = €

€

3 7

3 6

Question C - Division

€

C. 32

36

€

16 2

6 €

€

4 2

6

€

€

2 2

3

Simplify each radical.

Take square roots.

Simplify as a fraction (if possible).

Question D - Division

€

D. 5 + 50

5

€

€

5 + 25 2

5€

€

5 + 5 2

5

€

€

5

5+

5 2

5

€

€

1+ 2