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# The Basics

In people’s lives, the most numbers they use are rational numbers or numbers that have limits. But since the discovery of P or 3.1415926…, the world of irrational numbers or numbers that have no limits are presented in front of us. In current days, we represent irrational by using the radical signs or Ö . For all real numbers a and b, and all positive integers n, an= b, then a is the nth root of b. A positive nth root of b is written nÖ b and a negative nth root of b is written -nÖ b. To calculate radicals, you need a calculator for the most of times except those perfect squares or perfect triples and so on. Here are some additional properties of radicals:

For each real number a and each positive integer n, if n is odd, then nÖ a^n = a; if n is even, then nÖ a^n = |a|

The Multiplication Property of Radicals: For all real numbers a and b for which the radicals are defined and all positive integers n, nÖ ab = nÖ a x nÖ b.

The Division Property of Radicals: For all real numbers a and b, b ¹ 0, for which the radicals are defined, and for each positive integer n, nÖ (a/b) =nÖ a / nÖ b.

For all positive integers m and n, and all real numbers a for which the radical represents a real number, a m/n = (nÖ a)m = (nÖ am).

When a radical contains variables, then it is called a radical equation. The radical equations are usually solve by completing the squares or triples and so on. The most useful radical equation in the world of algebra is the Distance Formula that is used on the coordinate plane. It states that for points PI(X1, Y1) and P2(X2, Y2), P1P2 = Ö (X2 – X1)^2 + (Y2 – Y1)^2,. If the coordinates of point A are (X1, Y1) and the coordinates of point B are(X2, Y2), then the coordinates of the midpoint of AB are {(X1 + X2)/2, (Y1+Y2)/2}.

Radicals are very common in math and scientific researches.   Although we can solve most of the radicals by using a calculator today, it's still very important to understand the concepts.   Please take a few minutes break and click here to continue to the next chapter. (You can also click on the drop-down list below to jump to any chapter you like.)

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