Exponents & Radicals – Rules for Radicals

A radical is simply a root. The root taken is also referred to as an index. If you have a square root the index is two. If you have a cube root the index is three.

Exponents & Radicals – Rules for Radicals

A radical is simply a root. The root taken is also referred to as an index. If you have a square root the index is two. If you have a cube root the index is three.

For now we will work only with square roots.

Exponents & Radicals – Rules for Radicals

A radical is simply a root. The root taken is also referred to as an index. If you have a square root the index is two. If you have a cube root the index is three.

For now we will work only with square roots.

Perfect Squares – numbers that are gotten by multiplying the SAME number by itself. You could create a table of perfect squares…

36666

25555

16444

9333

4222

1111

2

2

2

2

2

2

144121212

121111111

100101010

81999

64888

49777

2

2

2

2

2

2

Exponents & Radicals – Rules for Radicals

Perfect Squares – numbers that are gotten by multiplying the SAME number by itself. You could create a table of perfect squares…

36666

25555

16444

9333

4222

1111

2

2

2

2

2

2

144121212

121111111

100101010

81999

64888

49777

2

2

2

2

2

2

To find a perfect square root, we will think of the perfect square table. Realize that table keeps going, I only went up to 12….

EXAMPLE : 25

Exponents & Radicals – Rules for Radicals

Perfect Squares – numbers that are gotten by multiplying the SAME number by itself. You could create a table of perfect squares…

36666

25555

16444

9333

4222

1111

2

2

2

2

2

2

144121212

121111111

100101010

81999

64888

49777

2

2

2

2

2

2

To find a perfect square root, we will think of the perfect square table. realize that table keeps going, I only went up to 12….

EXAMPLE : 25 What number did I multiply to get 25 ?

Exponents & Radicals – Rules for Radicals

Perfect Squares – numbers that are gotten by multiplying the SAME number by itself. You could create a table of perfect squares…

36666

25555

16444

9333

4222

1111

2

2

2

2

2

2

144121212

121111111

100101010

81999

64888

49777

2

2

2

2

2

2

To find a perfect square root, we will think of the perfect square table. realize that table keeps going, I only went up to 12….

EXAMPLE : 525 Notice there is no longer a radical over the answer…

Exponents & Radicals – Rules for Radicals

Perfect Squares – numbers that are gotten by multiplying the SAME number by itself. You could create a table of perfect squares…

36666

25555

16444

9333

4222

1111

2

2

2

2

2

2

144121212

121111111

100101010

81999

64888

49777

2

2

2

2

2

2

To find a perfect square root, we will think of the perfect square table. realize that table keeps going, I only went up to 12….

EXAMPLE : ?100 What number did I multiply to get 100 ?

Exponents & Radicals – Rules for Radicals

Perfect Squares – numbers that are gotten by multiplying the SAME number by itself. You could create a table of perfect squares…

36666

25555

16444

9333

4222

1111

2

2

2

2

2

2

144121212

121111111

100101010

81999

64888

49777

2

2

2

2

2

2

To find a perfect square root, we will think of the perfect square table. realize that table keeps going, I only went up to 12….

EXAMPLE : 10100 Notice there is no longer a radical over the answer…

Exponents & Radicals – Rules for Radicals

So now we could create a square root table…

12144 636

11121 525

10100 416

981 39

864 24

749 11

Exponents & Radicals – Rules for Radicals

Some rules for radicals :

1. A number / variable, divided by its square root, equals its square root

66

36 5

5

5 a

a

a

Exponents & Radicals – Rules for Radicals

Some rules for radicals :

1. A number / variable, divided by its square root, equals its square root

2. Numbers / variables under THE SAME root / index can be multiplied or divided

66

36 5

5

5 a

a

a

1052

172

34

Exponents & Radicals – Rules for Radicals

Some rules for radicals :

1. A number / variable, divided by its square root, equals its square root

2. Numbers / variables under THE SAME root / index can be multiplied or divided

66

36 5

5

5 a

a

a

1052

172

34

This rule also works in the OTHER direction, and is sometimes used to help simplify radicals…

3412

53

15

3

15

Exponents & Radicals – Rules for Radicals

Some rules for radicals :

3. Radicals act like variables. They can be added / subtracted as long as the root / index is the same AND what is under the radical is exactly the same.

787627672

Exponents & Radicals – Rules for Radicals

Some rules for radicals :

3. Radicals act like variables. They can be added / subtracted as long as the root / index is the same AND what is under the radical is exactly the same.

787627672

xxxx 86262 Just like

Exponents & Radicals – Rules for Radicals

Some rules for radicals :

3. Radicals act like variables. They can be added / subtracted as long as the root / index is the same AND what is under the radical is exactly the same.

787627672

xxxx 86262 Just like

aaa 1385

Exponents & Radicals – Rules for Radicals

Some rules for radicals :

3. Radicals act like variables. They can be added / subtracted as long as the root / index is the same AND what is under the radical is exactly the same.

787627672

xxxx 86262 Just like

aaa 1385

3724

You CAN NOT combine these. The number under the root is NOT the same…

Exponents & Radicals – Rules for Radicals

Simplifying radicals – sometimes a number under a radical can be broken into a perfect square part and a radical part.

Exponents & Radicals – Rules for Radicals

Simplifying radicals – sometimes a number under a radical can be broken into a perfect square part and a radical part.

EXAMPLE # 1 : Simplify 20

Exponents & Radicals – Rules for Radicals

Simplifying radicals – sometimes a number under a radical can be broken into a perfect square part and a radical part.

EXAMPLE # 1 : Simplify 20

54

102

201

Factors of 20

Method # 1 : 1st – list the factors of 20

Exponents & Radicals – Rules for Radicals

Simplifying radicals – sometimes a number under a radical can be broken into a perfect square part and a radical part.

EXAMPLE # 1 : Simplify 20

54

102

201

Factors of 20

Method # 1 : 1st – list the factors of 20

2nd – find the pair that has the biggest perfect square

Exponents & Radicals – Rules for Radicals

Simplifying radicals – sometimes a number under a radical can be broken into a perfect square part and a radical part.

EXAMPLE # 1 : Simplify 20

54

102

201

Factors of 20

Method # 1 : 1st – list the factors of 20

2nd – find the pair that has the biggest perfect square

3rd - break up your original number into this pair under radicals

5 4

Exponents & Radicals – Rules for Radicals

Simplifying radicals – sometimes a number under a radical can be broken into a perfect square part and a radical part.

EXAMPLE # 1 : Simplify 20

54

102

201

Factors of 20

Method # 1 : 1st – list the factors of 20

2nd – find the pair that has the biggest perfect square

3rd – break up the original number into that pair under radicals

4th – simplify the square root of the perfect square

52

5 4

Exponents & Radicals – Rules for Radicals

Simplifying radicals – sometimes a number under a radical can be broken into a perfect square part and a radical part.

EXAMPLE # 2 : Simplify 80

Method # 2 : Go thru the perfect square table and find the LARGEST perfect square that divides your number without a remainder.

We never use 1…

Exponents & Radicals – Rules for Radicals

Simplifying radicals – sometimes a number under a radical can be broken into a perfect square part and a radical part.

EXAMPLE # 2 : Simplify 80

Method # 2 : Go thru the perfect square table and find the LARGEST perfect square that divides your number without a remainder.

We never use 1…

63.14980

22.23680

2.32580

51680

89.8980

20480

When your answer falls below 2 stop.

Exponents & Radicals – Rules for Radicals

Simplifying radicals – sometimes a number under a radical can be broken into a perfect square part and a radical part.

EXAMPLE # 2 : Simplify 80

Method # 2 : Go thru the perfect square table and find the LARGEST perfect square that divides your number without a remainder.

We never use 1…

63.14980

22.23680

2.32580

51680

89.8980

20480

When your answer falls below 2 stop.

Even though there are two that work, we want the LARGEST perfect square…

Exponents & Radicals – Rules for Radicals

Simplifying radicals – sometimes a number under a radical can be broken into a perfect square part and a radical part.

EXAMPLE # 2 : Simplify 80

Method # 2 : Go thru the perfect square table and find the LARGEST perfect square that divides your number without a remainder.

We never use 1…

63.14980

22.23680

2.32580

51680

89.8980

20480

Break up your number into the perfect square that worked and the result…

5 16

Exponents & Radicals – Rules for Radicals

Simplifying radicals – sometimes a number under a radical can be broken into a perfect square part and a radical part.

EXAMPLE # 2 : Simplify 80

Method # 2 : Go thru the perfect square table and find the LARGEST perfect square that divides your number without a remainder.

We never use 1…

63.14980

22.23680

2.32580

51680

89.8980

20480

Reduce your perfect square…

5 16

54