1. ________
Find the sides of the following squares given the area of each.
Hints: How is the area of a square found?
multiply the sides
What is true about the sides of a square?
they are equal
Therefore the sides must be the same and multiply to = the area.

2. By finding the sides of the squares you found the____ ___ of the area.
x x3
By finding the sides of the squares you found the____ ___ of the area.
A = 400
A = 9x6
A = 4x2
A= 25

3. Square Roots
x x3
By finding the sides of the squares you found the square roots of the areas.
Square root -
A = 400
A = 9x6
A = 4x2
A= 25

4. Square Roots
A = 400
x x3
By finding the sides of the squares you found the square roots of the areas.
Square root - the side of a square
- a #/term that multiplies by itself to equal a certain #/term
A = 9x6
A= 25
A = 4x2

5. Square Roots What are the square roots of 16? 4 since (4)(4)=16
and -4 since (-4)(-4)=16
Therefore any positive # has 2 square roots, a positive and a negative.
The positive root is known as the principal root.

6. Square Roots √ asks for the positive root. Ex. √36 = 6
-√ asks for the negative root. Ex. -√64 = -8
+√ asks for both roots. Ex. + √81 = + 9
Evaluate.
a) √ b) -√ c)+√25x2 d) √-36

7. Square Roots Evaluate. √49 b) -√100 c)+√25x2 d) √-36
= = = + 5x = n.p
A negative # can not
have a square root
since no # times itself =
a neg.

8. Square Roots The √ symbol is called a radical.
The term under the radical is called the radicand. Ex. √25
radical radicand

9. Square Roots
Numbers such as 1, 4, 9, 16, 25, 36….. are known as _______ ________

10. Square Roots
Numbers such as 1, 4, 9, 16, 25, 36….. are known as square #s or perfect squares.
Square # - a # that has a whole # as a square root

11. Square Root Investigation # 1 Evaluate the following.
= 4
2.a) √10
=3.16
3.a) √256
=16
4.a) √81 =9
5.a) √12
=3.46
b) √4 √4
=2∙2 =4
b) √2 √5
=1.41∙2.24
=3.16
b) √16 √16
=4∙4
=16
b) √9 √9
= 3∙3
= 9
b) √4 √3
=2∙1.73
=3.46

12. What is true about (a) and (b) in each?
equal
What would be true about √60 and √20 √3 ?
How else could you write √50 ? √28 ?
ex. √50 = √25 √2
√28 = √4 √7

13. √ Conclusion # 1
The √ of a # is equal to the √ ’ s of its factors multiplied (or vice versa).
Ex. √ √ √99
= √4 √ = √9 √ = √9 √11
√3 √ √4 √ √ 16 √5
=√ = √ = √80

14. Use the above conclusion to simplify the following
Use the above conclusion to simplify the following. Write the expression in an equivalent, but different way. Don’t evaluate and express as a decimal approximation.
(Hint: think of the conclusion just made and express the √ in a different way)
a) √8
= √4 √2
= 2√2
2√2 is considered ‘simpler’ than √8 since the radicand in 2√2 is smaller than in √8

15. b) √27
= √9 √3
= 3 √3
c) √150
= √25 √6
= 5 √6

16. 3 has no perfect square factors (other than 1)
The square root of a non-perfect square is in simplest form/simplest radical form when the radicand does not have any factors that are perfect squares (other than1).
Ex. √12
= √4 √3
=2 √3
3 has no perfect square factors (other than 1) √6
already in simplest form since 6 has no perfect square factors

17. To simplify a radical:
-factor the radicand (try to factor so that you get a perfect square factor)
ex. √20
= √4 √5
-find the roots of the perfect square factors and leave the non-perfect squares as √
√20
= 2 √5
-continue until the √ is in simplest form (radicand does not contain any factors that are perfect squares)
2 √ does not contain any factors that are perfect squares

18. Simplify.
a) √ b) √ c) √ d) √10

19. √32
= √16 √2
= 4 √2
b) √300
= √25 √12
= 5 √12
= 5 √4 √3
= 5 ∙ 2 √3
= 10 √3
c) √99
= √9 √11
= 3 √11
d) √10
can not be simplified further since 10 has no factors that are perfect squares

20. √ Investigation # 2 Evaluate the following.
=2(3) + 3(3)
=6 + 9
=15
2.a) 5 √ √16
=5(4) + 6(4) =
=44
3.a) -4 √ √25
= -4(5) + 2(5) =
= -10
4.a) 7 √4 - 3 √4
=7(2) – 3(2)
=14 – 6
= 8
5.a) 10 √ √100
= 10(10) – 9(10)
= 100 – 90
= 10
6.a) 2 √4 + 3 √16
= 2(2) + 3(4) =
= 16
b) 5 √9
=5(3)
=15
b) 11 √16
=11(4)
=44
b) -2 √25
= -2(5)
= -10
b) 4 √4
= 4(2)
= 8
b) 1 √100
=1(10)
= 10
b) 5 √20
=5(4..5)
=22.5

21. What is true about the (a) and (b) in each?
equal, except # 6 where the radicands were different
What would be true about 5 √ √25 and 8 √25 ?
equal
What would be true about 12 √4 - 7 √4 and 5 √4 ?
Could 3 √ √10 be expressed in a simpler way ?
13 √10
Could 9 √ √7 be simplified ?
-6 √7

22. √ Conclusion # 2
An expression consisting of a sum or difference of radicals with the same radicands can be simplified by adding or subtracting the numbers in front of the radicals/the coefficients and keeping the radicand the same.
Ex. a) 8 √3 + 4 √3 b) 2 √4 + 3 √ c) 10 √ √25
=12 √ = 5 √ = 3 √25
= 5 (2) = 3(5)
= = 15
2 √3 + 3 √5
not possible since the radicands are not the same

23. Simplify the individual radicals first = 5 √9 √5 - 3 √4 √5
can’t be simplified as is because the radicands are different, but √8 can be simplified
= 2 √2 + 3 √4 √2
= 2 √2 + 3 (2) √2
= 2 √2 + 6 √2
= 8 √2
f) 5 √ √20
Simplify the individual radicals first
= 5 √9 √5 - 3 √4 √5
= 5(3) √5 – 3(2) √5
= 15 √5 - 6 √5
= 9 √5
g) 6 √ √44
= 6 √9 √ √4 √11
= 6(3) √ (2) √11
= 18 √ √11
= 38 √11