1. RADICAL EXPRESSIONS

2. ARE THE SAME AS SQUARE ROOTS
RADICAL EXPRESSIONS
ARE THE SAME AS SQUARE ROOTS

3. Radical Expressions Square Roots & Perfect Squares
Simplifying Radicals
Multiplying Radicals
Dividing Radicals
Adding Radicals

4. What is the square root? 64

5. The square root of a number is a # when multiplied by itself equals the number64 8 8

6. What is the square root? X2

7. A Square Root is a term when squared is the Perfect SquareX2 X X

8. What is the square root?
64x2

9. Answer
64x2 8x 8x

10. What is the square root?
4x2

11. Answer
4x2 2x 2x

12. What is the square root?
(x+3)2

13. Answer
(x+3)
(x+3)2
(x+3)

14. Radical Sign • This is the symbol for square root
• If the number 3 is on top of the v, it is called the 3rd root.

15. Radical Sign • This is the 4th root. • This is the 6th root
• If n is on top of the v, it is called the nth root.

16. Cube Root
If the sides of a cube are 3 inches Then the volume of the cube is 3 times 3 times 3 or 33 which is 27 cubic inches.

17. NUMBER Perfect Squares
Numbers that are perfect squares are:
12=1, 22= 4, 32= 9, 42= 16, 52= 25, 62= 36, 72= 49, = 64, 92= 81, 102= 100, …

18. Recognizing Perfect Squares (NAME THE SQUARE ROOTS)
Variables that are perfect squares are:
x2, a4, y22, x100…
(Any even powered variable is a perfect square)
x25
EVEN POWERED
EXPONENTS ARE
SQUARES
x25
x50

19. NAME THE SQUARE ROOT
9x50

20. 3x25
EVEN POWERED
EXPONENTS ARE
SQUARES
3x25
9x50

21. Recognizing Perfect Squares (NAME THE SQUARE ROOTS)x2 4
25x2
4x6

22. Recognizing Perfect Squares (NAME THE SQUARE ROOTS)x 2 x x2 2 4 5x
2x3 5x
25x2
2x3
4x6

23. Simplifying Square Roots

24. Simplifying Square Roots
We can simplify if 8 contains a Perfect Square

25. Simplifying Square Roots
Look to factor perfect squares (4, 9, 16, 25, 36…)
Put perfect squares in first radical and the other factor in 2nd.
Take square root of first radical

26. Simplifying Square Roots

27. Simplifying Square Roots
1. Factor any perfect squares (4, 9, 16, 25, x2, y6…)
(Put perfect squares in first radical and other factor in 2nd)

28. Simplifying Square Roots
1. Factor any perfect squares (4, 9, 16, 25, x2, y6…)
(& Put perfect squares in first radical and other factor in 2nd)
2. Take square root of first radical

29. Simplifying Square Roots
1. Factor any perfect squares (4, 9, 16, 25, x2, y6…)
(& Put perfect squares in first radical and other factor in 2nd)
2. Take square root of first radical

30. Simplifying Square Roots
1. Factor any perfect squares (4, 9, 16, 25, x2, y6…)
(& Put perfect squares in first radical and other factor in 2nd)
2. Take square root of first radical

31. Simplifying Square Roots

32. Simplifying Square Roots
1. Factor any perfect squares (4, 9, 16, 25, 36…)
(& Put perfect squares in first radical and other factor in 2nd)

33. Simplifying Square Roots
1. Factor any perfect squares (4, 9, 16, 25, 36…)
(& Put perfect squares in first radical and other factor in 2nd)
2. Take square root of first radical

34. Simplifying Square Roots
1. Factor any perfect squares (4, 9, 16, 25, 36…)
(& Put perfect squares in first radical and other factor in 2nd)

35. Simplifying Radicals 1. Factor any perfect squares (4, 9, 16, 25, 36…)
(& Put perfect squares in first radical and other factor in 2nd)

36. Simplifying Radicals
1. Factor any perfect squares (4, 9, 16, 25, x2, y6…)
(& Put perfect squares in first radical and other factor in 2nd)
2. Take square root of first radical

37. Simplifying Fraction Radicals

38. Simplifying Fraction Radicals
1. Factor any perfect squares (4, 9, 16, 25, x2, y6…) from NUMERATOR & DENOM.
(& Put perfect squares in first radical and other factor in 2nd)

39. Simplifying Fraction Radicals
1. Factor any perfect squares (4, 9, 16, 25, x2, y6…) from NUMERATOR & DENOM.
(& Put perfect squares in first radical and other factor in 2nd)
2. Take square root of first radical

40. Simplifying Fraction Radicals
1. Factor any perfect squares (4, 9, 16, 25, x2, y6…) from NUMERATOR & DENOM.
(& Put perfect squares in first radical and other factor in 2nd)
2. Take square root of first radical
3. REDUCE

41. Finding Perfect Squares
The most common perfect squares are 4 & 9.
USE DIVISIBILITY RULES:
A number is divisible by 4 if the last 2 digits are divisible by 4.
( 23,732 is divisible by 4 since 32 is.)
A number is divisible by 9 if the sum of it’s digits are.
(4653 is divisible by 9 since the sum of it’s digits are.)

42. Finding Perfect Squares
MORE EASY TO FIND PERFECT SQUARES:
A number with an even number of Zeros.
( 31,300 is divisible by 10, 70,000 is divisible by 100)
A number ending in 25, 50 or 75 is divisible by 25.
(425 & 350 & 775 are all divisible by 25)

43. Perfect Squares Guide Divisibility Rules for Perfect Squares
4: If Last 2 Digits are Divisible by 4
9: If Sum of Digits are Divisible by 9
16: Use 4 Rule
25: If # ends in 25, 50, 75 or 00
36: Use 4 or 9 Rule
49: Look for multiple of 50 and subtract multiple
64: Use 4 Rule
81: Use 9 Rule
100: If # ends in 00
121: No Rule, Divide # by 121
144: Use 4 or 9 Rule

45. Multiply Radicals

46. Multiply Radicals
Multiply to one radical

47. Multiply Radicals
Multiply to one radical

48. Multiply Radicals Multiply to one radical Simplify
• Factor any perfect squares (4, 9, 16, x2, y6…)

49. Multiply Radicals Multiply to one radical Simplify
• Factor any perfect squares (4, 9, 16, x2, y6…)

50. Multiply Radicals Multiply to one radical Simplify
• Factor any perfect squares (4, 9, 16, x2, y6…)
• Take square root of first radical

51. Multiply Radicals

52. Multiply Radicals
Multiply to one radical

53. Multiply Radicals
Multiply to one radical

54. Multiply Radicals Multiply to one radical Simplify
• Factor any perfect squares (4, 9, 16, x2, y6…)

55. Multiply Radicals Multiply to one radical Simplify
• Factor any perfect squares (4, 9, 16, x2, y6…)
• Take square root of first radical

56. Multiply Radicals

57. Multiply Radicals
Multiply to one radical

58. Multiply Radicals
Multiply to one radical

59. Multiply Radicals Multiply to one radical Simplify
• Factor any perfect squares (4, 9, 16, x2, y6…)

60. Multiply Radicals Multiply to one radical Simplify
• Factor any perfect squares (4, 9, 16, x2, y6…)
• Take square root of first radical

61. Multiply Radicals
Multiply to one radical

62. Multiply Radicals
Multiply to one radical

63. Multiply Radicals Multiply to one radical Simplify
• Factor any perfect squares (4, 9, 16, x2, y6…)

64. Multiply Radicals Multiply to one radical Simplify
• Factor any perfect squares (4, 9, 16, x2, y6…)
• Take square root of first radical

65. Another Way to Multiply Radicals

66. Multiply Radicals By Factoring Perfect Squares First

67. Multiply Radicals By Factoring Perfect Squares First
Factor any perfect squares

68. Multiply Radicals By Factoring Perfect Squares First
Factor any perfect squares

69. Multiply Radicals By Factoring Perfect Squares First
Factor any perfect squares
Take square roots of perfect squares

70. Multiply Radicals By Factoring Perfect Squares First
Factor any perfect squares
Take square roots of perfect squares
Multiply #,s &

71. Multiply Radicals with Distributive Prop.

72. Multiply Radicals with Distributive Prop.
Mult. With Distr. Property

73. Multiply Radicals with Distributive Prop.
Mult. With Distr. Property

74. Multiply Radicals with Distributive Prop.
Mult. With Distr. Property
Factor any perfect squares

75. Multiply Radicals with Distributive Prop.
Mult. With Distr. Property
Factor any perfect squares
Take square roots of perfect squares
Simplify if possible

76. Multiply Radicals with Distributive Prop.
Multiply Terms (Use Multiplying Boxes & CLT)
Simplify if possible x 3x
3x2

77. Dividing Square Roots
1. Factor any perfect squares (4, 9, 16, 25, x2, y6…)from the numerator & denominator
(& Put perfect squares in first radical and other factor in 2nd)
2. Take square root of first radical.
3. Reduce

78. Dividing Square Roots

79. Dividing Square Roots
1. Factor any perfect squares (4, 9, 16, 25, x2, y6…)from the numerator & denominator
(& Put perfect squares in first radical and other factor in 2nd)

80. Dividing Square Roots
1. Factor any perfect squares (4, 9, 16, 25, x2, y6…)from the numerator & denominator
(& Put perfect squares in first radical and other factor in 2nd)

81. Dividing Square Roots
1. Factor any perfect squares (4, 9, 16, 25, x2, y6…)from the numerator & denominator
(& Put perfect squares in first radical and other factor in 2nd)
2. Take square root of first radical.

82. Dividing Square Roots
1. Factor any perfect squares (4, 9, 16, 25, x2, y6…)from the numerator & denominator
(& Put perfect squares in first radical and other factor in 2nd)
2. Take square root of first radical.
3. Reduce

83. Divide Radicals

84. Divide Radicals
Divide

85. Divide Radicals
Divide

86. Divide Radicals
Divide
Simplify

87. Divide Radicals
Divide
Simplify

88. Divide Radicals Rationalizing the Denominator

89. Divide Radicals Rationalizing the Denominator
Mult. Numerator & Denom. By Denom. to get a Perfect Square in Denominator

90. Divide Radicals Rationalizing the Denominator
Mult. Numerator & Denom. By Denom. to get a Perfect Square in Denominator

91. Divide Radicals Rationalizing the Denominator
Mult. Numerator & Denom. By Denom. to get a Perfect Square in Denominator
Take the square root of the Perfect Square
Simplify

92. Divide Radicals Rationalizing the Denominator
Mult. Numerator & Denom. By Denom. to get a Perfect Square in Denominator
Take the square root of the Perfect Square
Simplify

93. Divide Radicals Rationalizing the Denominator(2 term)
Rationalizing the denominator means making the denominator rational or REMOVING IRRATIONAL TERMS

94. Divide Radicals Rationalizing the Denominator(2 term)
Mult. Numerator & Denom. By Conjugate to get a Perfect Square in Denominator
The conjugate of a binomial is SWITCHING THE SIGN BETWEEN THEM

95. Divide Radicals Rationalizing the Denominator(2 term)
Mult. Numerator & Denom. By Conjugate to get a Perfect Square in Denominator 8 8 64

96. Divide Radicals Rationalizing the Denominator(2 term)
Mult. Numerator & Denom. By Conjugate to get a Perfect Square in Denominator
Take the square root of the Perfect Square

97. Divide Radicals Rationalizing the Denominator(2 term)
Mult. Numerator & Denom. By Conjugate to get a Perfect Square in Denominator
Take the square root of the Perfect Square
Simplify

98. Square Roots of Perfect Squares
1. Factor any perfect squares
2. Take square root of perfect square.

99. Values that make it Real
1. Set term in sq. root ≥ 0
2. Solve

100. Values that make it Real

101. Values that make it Real
1. Set term in sq. root ≥ 0
2. Solve

102. Values that make it Real
Just note that if x is any real number (all numbers on the number line)
x2 can never be negative, so
(x2+16) will always be positive.
The answer is any real number

103. Negative Square Roots
Can you take the square root of a negative number?

104. Can you take the square root of a negative number?NO
Because a negative number squared is positive

105. Simplifying Square Roots
1. Factor any perfect squares (4, 9, 16, 25, 36…)
(& Put perfect squares in first radical and other factor in 2nd)
2. Take square root of first radical

106. Simplifying Negative Square Roots
1. Factor any perfect squares (4, 9, 16, 25, 36…)
(& Put perfect squares in first radical and other factor in 2nd)
2. Take square root of first radical
3. The square root of -1 is called IMAGINARY
(We will review this later)

107. Adding Radicals

108. Adding Radicals
Same as Adding
LIKE TERMS

109. As Simple As

110. As Simple As + =

111. Similar Radical's
are Like Terms + =

112. Can't Combine
Unlike Terms + = +
a + b = a b

113. Only Combine
Like Terms + = + +
a + a + b = 2a + b

114. Always Combine
Like Terms + =
a + a = a

115. Always Combine
Like Terms + =
a + a = a

116. As Simple As

117. As Simple As

118. one radical x equals two radical x's
As Simple As
one radical x plus
one radical x equals two radical x's

119. Same as Adding
LIKE TERMS

120. Same as Adding
LIKE TERMS

121. Same as Adding
LIKE TERMS

122. Same as Adding
LIKE TERMS

123. Same as Adding
LIKE TERMS

124. Same as Adding
LIKE TERMS

125. Can you add
unlike terms?

126. NO
You can't combine
unlike terms

127. Try This One

128. Answer

129. Write Down Your
Answers

130. Answers

131. Write Down Your
Answers

132. Answers

133. Adding Radicals

134. Adding Radicals
Simplify to Like

135. Adding Radicals
Simplify to Like

136. Adding Radicals
Simplify to Like
Add the Like

137. Adding Radicals

138. Adding Radicals
Simplify to Like • GCF

139. Adding Radicals
Simplify to Like • GCF

140. Adding Radicals
Simplify to Like • GCF •

141. Adding Radicals
Simplify to Like • GCF • •Simplify

142. Adding Radicals
Simplify to Like • GCF • •Simplify
Add the Like Terms

143. Adding Radicals

144. Adding Radicals
Simplify to Like • Rationalize Denominator

145. Adding Radicals
Simplify to Like • Rationalize Denominator

146. Adding Radicals
Simplify to Like • Rationalize Denominator
Add the Like Terms