Friday, March 9, 2007

my table full of numbers


THIS POST IS ABOUT RADICALS

this is not a root of a tree. but a root of a number.
does it sound scary? ofcourse it does. first impression to radicals last,NOT. you may think alot of numbers are seen in your paper or the entire table rather, with all those papers all over the place containtg numbers. worse is that zero is there..XD

i- is what you call an imaginary number.
-it is the square root of -1 [fascinates me alot]

i cube = - square root of -1 [same same]

so let me tell you a story about the radicals:

Roots and Radicals deserve their own chapter and homework because they occur frequently in applications.

Let $ n\<span class=geq 2 $" align="middle" border="0" height="39" width="57"> be a natural number, and let $ a $ be a real number. The $ n $-th root of $ a $ is a number $ b $ that satisfies $ a = b^n. $ The number $ b $ is denoted by

$\<span class=displaystyle b = \root n \of a. $" align="middle" border="0" height="48" width="81">

For example, $ \root 4 \of 81 = 3 $ since $ 3^4 = 81 $, and $ \root 5 \of 32 = 2 $ since $ 2^5= 32 $.

The symbol $ \<span class=sqrt{\phantom{yz}} $" align="middle" border="0" height="42" width="45"> is called the radical symbol, and an expression involving it is called a radical (expression).

If $ n=2 $ then $ b $ is the square root of $ a $ and the number $ 2 $ is usually omitted. For example, $ \root 2 \of 25 = \<span class=sqrt{25} = 5. $" align="middle" border="0" height="50" width="155">

If $ n=3 $, then $ b $ is the cube root of $ a $. For example, the cube root of $ 27 $ is $ 3 $, and that of $ 8 $ is $ 2 $.

If $ n $ is even and $ a $ is positive then there are two $ n $-th roots of $ a $, each being the negative of the other. For example, since $ 5^2 = (-5)^2 = 25 $ there are two square roots of $ 25 $. In that case by convention the symbol $ \root n \of a $ means the positive $ n $-th root of $ a $, and it is called the principal ($ n $-th) root of $ a $.

If $ a $ is negative and $ n $ is odd then there is just one $ n $-th root, and it is negative also. For example, $ \root 3 \of -1 = \root 5 \of -1 = -1. $

At this stage we do not know of an $ n $-th root if $ n $ is even and $ a $ is negative. This leads to the subject of complex numbers which we will take up later in the course.

Radicals are just special cases of powers, and you can simplify much of your thinking by keeping this fact in mind:

$\<span class=

It follows immediately from that observation and the properties of powers that

$\<span class=>

Solving Radical Equations

An equation involving radicals is called a radical equation (naturally). To solve it you simply apply our general principle:

To solve an equation figure out what bothers you and then do the same thing on both sides of the equation to get rid of it.

To get rid of a radical you take it to a power that will change the rational exponent to a natural number. This will work if the radical is on one side of the equation by itself.


Hope you enjoyed the story about the radicals and roots. because you will never get the answer if you will not try to analyze it. numbers might be scary on the outside but meaningful on the inside.


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