Radicals

1
Radicals

• Square roots when left under the square root or
  radical sign are referred to as radicals.
• They are separate class of numbers like whole
  numbers or fractions and have certain properties
  in common.
• If I asked what 42 was equal to, you might think
  4 x 4 16 duh!
• Then if I asked you what the v16 was equal to,
  its 4
• Now if I ask you what v16 x v16 is equal to,
  its 16
• The same as 4 x 4 , v16 x v16.
• What about v7 x v7 , 7 of course
• Now what about v7 x v5, what is it? v35

2
Radicals

• So if v5 x v5 v25 5 and v7 x v7 v49 7
• Then v5 x v7 v35 and v35 x v5 v165
• BUT v5 x v5 x v7 5 x v7 5v7 right
• Any time you have a number like v288 we can start
  factoring out radical factors like v2
• For example v288 v2 x v144
• Then sometimes instead of factoring out v2s and
  v3s
• We can see that v144 v12 x v12 12
• Simply put, the square root of 144 is 12
• Anytime we have a pair of vxs they can be
  factored out as an x. Look at some examples

3
Radicals

• Lets start backwards
• v3 x v3 x v5 x v2 x v7 x v5 x v2 v6300
• Use the rules of divisibility
• 6300 ends in 00, evenly divisible by 4
• v2 x v2 x v1575 or 2v1575
• Next I can see at least one 5 so 2 x v5 x v315
• Then 2 x v5 x v5 x v63 2 x 5 x v63 10v63
• Immediately I know v7 x v9 v7 x 3
• Now its 30v7
• What good is all this?

4
Radicals

• Remember the 3,4, 5 triangle
• A2 B2 C2
• 32 42 52 or 9 16
  25
• There are two other triangles that even more
  important in engineering, navigation, GPS and
  higher math.
• When I cut a square in half along the diagonal I
  get two identical isosceles right triangles

4
4
4
5
Pythagorean Triples

• Since the side came from a square both short legs
  are equal, making them isosceles
• The long side or hypotenuse can be learned using
  the Pythagorean theorem
• A2 B2 C2
• 42 42 C2
• 16 16 C2
• v 32 v C2
• v32 C
• BUT v32 v2 x v16 4v2
• REMEMBER ?

• .

4v2
4
4
6
Pythagorean Triples

• No matter what I do to the side of the square
• The third side of the triangle is going to be Sv2
• If the square is 5 on its side
• The diagonal is 5v2
• If the side is 52
• The diagonal is 52v2
• Even if the side IS v2
• The diagonal is v2 x v2 2
• Remember v7 x v7 7
• v29 x v29 29

• .

4v2
4
4
7
Pythagorean Triples

• Another really important triangle.
• Take an equilateral triangle and cut it in half
• The result is a 30 60 90 triangle
• This triangle has some powerful properties
• Whatever the short base, the hypotenuse is double
• Furthermore
• A2 B2 C2
• 12 B2 22
• 1 B2 4
• B2 3
• B v3

• .

6
3v3
3
8
Special Triangles

• These special examples of Pythagorean triples
  are known as Special Triangles
• They always maintain the same relationship to
  similar triangles
• For example 3, 3, 3v2 or
• 5, 5, 5v2
• AND
• 1, v3, 2 or 10, 10v3, 20
• or 7, 7v3, 14
• Dont be fooled
• v3 , 3, 2v3 ugly huh?
• The real trick in any instance is to multiply
  everything by either
• 1,1, v2 for Isosceles right
• or 1,2,v3 for 30-60-90 half of an equilateral.

• .

45
45
90
30
90
60