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 = 327

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 = 3a. 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:

xa/n = nxa = (nx)a

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

Proof: let's have nab that equals to (ab)1/n, which from the basic formula up of the exponent, comes to a 1/n.b1/n, or nanb

2nx ≥ 0 n - natural number(if x ≥ 0)
na/b = na/nb

Proof: na/b = (a/b)1/n and from the basic equations of the exponent, comes to a1/n/b1/n, or na/nb

nma = n . ma

Proof: if you have nma that equals to na1/m, which equals to (a1/m)1/n and from the basic equations of the exponent, comes to a1/(m.n), or n . ma

If 0 ≤ x < y then nx < ny
##### Graph of third root

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