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:
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.b1/n, or n√an√b
Proof: n√a/b = (a/b)1/n and from the basic equations of the exponent, comes to a1/n/b1/n, or n√a/n√b
Proof: if you have n√ that equals to n√, which equals to (a1/m)1/n and from the basic equations of the exponent, comes to a1/(m.n), or n . m√a
Graph of sqare root
Graph of third root
More about radicals in the maths forum
- Power Of 2
- Inequality 4
- Transform in a sum of simple radicals, please!
- Solve.., Find the roots of: z^3 + 6z^2 - 4z - 24 = 0