Radicals
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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 √16 was equal to, it’s 4
Now if I ask you what √16 x √16 is equal to, it’s 16
The same as 4 x 4 , √16 x √16.
What about √7 x √7 , 7 of course
Now what about √7 x √5, what is it? √35
Radicals
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So if √5 x √5 = √25 = 5 and √7 x √7 = √49 = 7
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Then √5 x √7 = √35 and √35 x √5 = √165
BUT √5 x √5 x √7 = 5 x √7 = 5√7 right
Any time you have a number like √288 we can start
factoring out radical factors like √2
For example √288 = √2 x √144
Then sometimes instead of factoring out √2’s and √3’s
We can see that √144 = √12 x √12 = 12
Simply put, the square root of 144 is 12
Anytime we have a pair of √x’s they can be factored
out as an “x”. Look at some examples
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Radicals
Let’s start backwards
 √3 x √3 x √5 x √2 x √7 x √5 x √2 = √6300
 Use the rules of divisibility
 6300 ends in 00, evenly divisible by 4
 √2 x √2 x √1575 or 2√1575
 Next I can see at least one 5 so 2 x √5 x √315
 Then 2 x √5 x √5 x √63 = 2 x 5 x √63 = 10√63
 Immediately I know √7 x √9 = √7 x 3
 Now it’s 30√7
 What good is all this?
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Radicals
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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,4 GPS and higher math.
When I cut a square in half along the diagonal I get
two identical isosceles right triangles
4
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Pythagorean Triples
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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 = C 2
16 + 16 = C2
√ 32
= √ C2
√32
= C
BUT √32 = √2 x √16 = 4√2
REMEMBER ?
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4√2
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Pythagorean Triples
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No matter what I do to the
side of the square
The third side of the triangle
is going to be S√2
If the square is 5 on its side
The diagonal is 5√2
If the side is 52
The diagonal is 52√2
Even if the side IS √2
The diagonal is √2 x √2 = 2
Remember √7 x √7 = 7
√29 x √29 = 29
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4√2
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Pythagorean Triples
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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 = √3
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6
6
3
3
30
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3√3
90
60
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Special Triangles
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These “special” examples of
Pythagorean triples are known as
Special Triangles
They always maintain the same
relationship to similar triangles
For example 3, 3, 3√2 or
5, 5, 5√2
AND
1, √3, 2 or 10, 10√3, 20
or 7, 7√3, 14
Don’t be fooled
√3 , 3, 2√3 ugly huh?
The real trick in any instance is to
multiply everything by either
1,1, √2 for Isosceles right
or 1,2,√3 for 30-60-90 half of an
equilateral.
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90
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30
90
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