Radicals
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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 4 2 was equal to, you might think 4 x 4 = 16 duh!

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Radicals

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Radicals

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


Radicals1

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


Radicals2

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

  • 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 ?

.

Pythagorean Triples

4√2

4

4


Pythagorean triples1

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

.

Pythagorean Triples

4√2

4

4


Pythagorean triples2

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

.

6

6

3

3

30

90

60

Pythagorean Triples

6

3√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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Special Triangles

45

90

45

30

90

60


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