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 actually 3 with exponent 2 and 3.

Basically what radical is, is a function which finds a divider, of the argument, which upped on exponent gives us the argument. Sometimes this divider is not a rational number. The radical is actually the opposite function of an exponent. It even can be write down with the help of an exponent.

So in our case the square(2nd) 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.

$\sqrt[2]{x}$ is $\sqrt{x}$

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:

$x^{\frac{m}{n}}=\sqrt[n]{x^m}=(\sqrt[n]{x})^m$

Radical Formulas

If n is odd:
$\sqrt[n]{x^n}=x$

If n is even:
$\sqrt[n]{x^n}=|x|$ - the absolute value of x

Example: $\sqrt[3]{x^3}=x$ but $\sqrt[4]{x^4}=|x|$

$\sqrt[n]{a \cdot b}=\sqrt[n]{a}\cdot \sqrt[n]{b}$

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

$\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$

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

$\sqrt[n]{\sqrt[m]{a}}=\sqrt[n\cdot m]{a}$

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

²ⁿ√x ≥ 0 n is a natural number and x ≥ 0

Radical monotony

If 0 ≤ x < y then ⁿ√x < ⁿ√y

Graph of square root

Graph of third root

More about radicals in the maths forum

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