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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

- So if √5 x √5 = √25 = 5 and √7 x √7 = √49 = 7
- 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

- 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?

- 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

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

√ 32 = √ C2

√32 = C

BUT √32 = √2 x √16 = 4√2

REMEMBER ?

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4√2

4

4

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

4

4

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

90

60

6

3√3

3

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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45

90

45

30

90

60