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Radical, What is Radical

Let us take the number 9. Nine divided by 3 equals to the divider 3 => 9/3 = 3, so 3.3 = 9 or 3² = 9. Let us take another number, 27 this time, 27 = 3.3.3 = 3³. So we found that 9 and 27 are actualy 3 with exponent 2 and 3. Basicly what radical is, is a fuction which finds a dvider, of the argument, which upped on exponent gives us the argument. Sometimes this divider is not a rational number. The radical is actualy the oposite function of an exponent. It even can be write down with the help of an exponent. So in our case the square(2-nd) root of 9 is 3, √9 and the third root of 27 is 3 = ³√27

If a is positive real number then the equation x² = a has two solutions: x = +√a or x = -√a.

If a is real number then the equation x³ = a has only one solution => x = ³√a. With the help of the equtions above we solve square and cubic equations. A root can be write down with the help of an exponent, the following rule applies:

x^a/n = ⁿ√x^a = (ⁿ√x)^a

Radical Formulas

These are the basic equations, which you need to remember.

Proof: let's have ⁿ√ab that equals to (ab)^1/n, which from the basic formula up of the exponent, comes to a^1/n.b^1/n, or ⁿ√aⁿ√b

²ⁿ√x ≥ 0 n - natural number(if x ≥ 0)

ⁿ√a/b = ⁿ√a/ⁿ√b

Proof: ⁿ√a/b = (a/b)^1/n and from the basic equations of the exponent, comes to a^1/n/b^1/n, or ⁿ√a/ⁿ√b

ⁿ√^m√a = ^{n . m}√a

Proof: if you have ⁿ√^m√a that equals to ⁿ√a^1/m, which equals to (a^1/m)^1/n and from the basic equations of the exponent, comes to a^1/(m.n), or ^{n . m}√a