Radicals l l l l Square roots when

Radicals l l l l 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 l So l l l l 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 √ 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 l l l l l Let’s start backwards √ 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 √ 1575 or 2√ 1575 Next I can see at least one 5 so 2 x √ 5 x √ 315 Then 2 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 l l l Remember the 3, 4, 5 triangle A 2 + B 2 = C 2 32 + 4 2 = 5 2 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 4

Pythagorean Triples l l l l l 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 A 2 + B 2 = C 2 42 + 4 2 = C 2 16 + 16 = C 2 √ 32 = √ C 2 √ 32 = C BUT √ 32 = √ 2 x √ 16 = 4√ 2 REMEMBER ? . 4√ 2 4 4

Pythagorean Triples l l l l l 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 . 4√ 2 4 4

Pythagorean Triples l l l 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 A 2 + B 2 = C 2 12 + B 2 = 2 2 1 + B 2 = 4 B 2 = 3 B = √ 3 . 6 6 3 3 30 6 3√ 3 90 60 3

Special Triangles l l l 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. . 45 90 45 30 90 60