1. 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.

2. 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.

3. 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…

4. 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…
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 :

5. 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…
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 :
What number did I multiply to get 25 ?

6. 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…
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 :
Notice there is no longer a radical over the answer…

7. 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…
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 :
What number did I multiply to get 100 ?

8. 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…
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 :
Notice there is no longer a radical over the answer…

9. Exponents & Radicals – Rules for Radicals
So now we could create a square root table…

10. Exponents & Radicals – Rules for Radicals
Some rules for radicals :
1. A number / variable, divided by its square root, equals its square root

11. 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

12. 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
This rule also works in the OTHER direction, and is sometimes used to help simplify radicals…

13. 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.

14. 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.
Just like

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.
Just like

16. 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.
Just like
You CAN NOT combine these. The number under the root is NOT the same…

17. 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.

18. 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

19. 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
Factors of 20
Method # 1 : 1st – list the factors of 20

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
Factors of 20
Method # 1 : 1st – list the factors of 20
2nd – find the pair that has the biggest perfect square

21. 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
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

22. 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
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

23. 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
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…

24. 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
When your answer falls below 2 stop.
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…

25. 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
Even though there are two that work, we want the LARGEST perfect square…
When your answer falls below 2 stop.
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…

26. 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
Break up your number into the perfect square that worked and the result…
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…

27. 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
Reduce your perfect square…
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…

28. Exponents & Radicals – Rules for Radicals
Square root expressions

29. Exponents & Radicals – Rules for Radicals
Square root expressions

30. Exponents & Radicals – Rules for Radicals
Square root expressions

31. Exponents & Radicals – Rules for Radicals
Square root expressions

32. Exponents & Radicals – Rules for Radicals
Square root expressions

33. Exponents & Radicals – Rules for Radicals
Square root expressions
EXAMPLE # 4 : For which value of x should the expression be further simplified ?
a) x = 15
b) x = 21
c) x = 39
d) x = 51

34. Exponents & Radicals – Rules for Radicals
Square root expressions
EXAMPLE # 4 : For which value of x should the expression be further simplified ?
a) x = 15
b) x = 21
c) x = 39
d) x = 51
To solve this, we need to write down the factors of the given numbers and find a match with the factors of the number in the expression…

35. Exponents & Radicals – Rules for Radicals
Square root expressions
EXAMPLE # 4 : For which value of x should the expression be further simplified ?
a) x = 15 3 x = 7 x 11
b) x = 21 3 x 7
c) x = 39 3 x 13
d) x = 51 3 x 17
To solve this, we need to write down the factors of the given numbers and find a match with the factors of the number in the expression…

36. Exponents & Radicals – Rules for Radicals
Square root expressions
EXAMPLE # 4 : For which value of x should the expression be further simplified ?
a) x = 15 3 x = 7 x 11
b) x = 21 3 x 7
c) x = 39 3 x 13
d) x = 51 3 x 17
To solve this, we need to write down the factors of the given numbers and find a match with the factors of the number in the expression…
As you can see, there is a match between 77 and 21 ( the factor 7 )

37. Exponents & Radicals – Rules for Radicals
Square root expressions
EXAMPLE # 4 : For which value of x should the expression be further simplified ?
a) x = 15 3 x = 7 x 11
b) x = 21 3 x 7
c) x = 39 3 x 13
d) x = 51 3 x 17
Now rewrite the expression with the factors of 77 and 21…

38. Exponents & Radicals – Rules for Radicals
Square root expressions

39. Exponents & Radicals – Rules for Radicals
Square root expressions

40. Exponents & Radicals – Rules for Radicals
Square root expressions

41. Exponents & Radicals – Rules for Radicals
Square root expressions word/application problems
EXAMPLE : The diagonal of a computer monitor is inches long. Find the
simplified form for the length of the diagonal measure.

42. Exponents & Radicals – Rules for Radicals
Square root expressions word/application problems
EXAMPLE : The diagonal of a computer monitor is inches long. Find the
simplified form for the length of the diagonal measure.
SOLUTION : = ∙ 10

43. Exponents & Radicals – Rules for Radicals
Square root expressions word/application problems
EXAMPLE : The diagonal of a computer monitor is inches long. Find the
simplified form for the length of the diagonal measure.
SOLUTION : = ∙ 10
160 = 4 ∙ 10

44. Exponents & Radicals – Rules for Radicals
Square root expressions word/application problems
EXAMPLE : The diagonal of a computer monitor is inches long. Find the
simplified form for the length of the diagonal measure.
SOLUTION : = ∙ 10
160 = 4 ∙ 10
ANSWER :

45. 𝐴=9𝑥 Exponents & Radicals – Rules for Radicals
Square root expressions word/application problems
EXAMPLE # 2 : The diagram shows a window in the shape of a square. Each
side of the square measures feet, and the area is equal
to 9𝑥 square feet. Find the value of 𝑥 .
𝐴=9𝑥
3 5 ft.

46. 𝐴=9𝑥 Exponents & Radicals – Rules for Radicals
Square root expressions word/application problems
EXAMPLE # 2 : The diagram shows a window in the shape of a square. Each
side of the square measures feet, and the area is equal
to 9𝑥 square feet. Find the value of 𝑥 .
Solution : 𝐴= 𝑠 2
𝐴=9𝑥
3 5 ft.

47. 𝐴=9𝑥 Exponents & Radicals – Rules for Radicals
Square root expressions word/application problems
EXAMPLE # 2 : The diagram shows a window in the shape of a square. Each
side of the square measures feet, and the area is equal
to 9𝑥 square feet. Find the value of 𝑥 .
Solution : 𝐴= 𝑠 2
9𝑥= (3 5 ) 2
𝐴=9𝑥
3 5 ft.

48. 𝐴=9𝑥 Exponents & Radicals – Rules for Radicals
Square root expressions word/application problems
EXAMPLE # 2 : The diagram shows a window in the shape of a square. Each
side of the square measures feet, and the area is equal
to 9𝑥 square feet. Find the value of 𝑥 .
Solution : 𝐴= 𝑠 2
9𝑥= (3 5 ) 2
9𝑥= (3) 2 ∙ ( 5 ) 2
9𝑥=9 ∙5
𝐴=9𝑥
3 5 ft.

49. 𝐴=9𝑥 Exponents & Radicals – Rules for Radicals
Square root expressions word/application problems
EXAMPLE # 2 : The diagram shows a window in the shape of a square. Each
side of the square measures feet, and the area is equal
to 9𝑥 square feet. Find the value of 𝑥 .
Solution : 𝐴= 𝑠 2
9𝑥= (3 5 ) 2
9𝑥= (3) 2 ∙ ( 5 ) 2
9𝑥=9 ∙5
9𝑥=45
𝑥=5
𝐴=9𝑥
3 5 ft.