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 √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

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

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?

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

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

√ 32 = √ C2

√32 = C BUT √32 = √2 x √16 = 4√2 REMEMBER ?

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4

4

4√2

Pythagorean Triples

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

4

4√2

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 = √3

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6 6

3 3

30

6090

3

6

3√3

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, 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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30

6090

45

4590