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Roots

The n-th root of a number is the value that can be multiplied by itself n times to give the original number.

Example: Square root

The square root of a number is the value that can be multiplied by itself to give the original number.

Square roots are usually denoted with a radical symbol or radix (from greek radix = root):

sqrt{a}

where a is a number.

For example, we want to find the square root of 9:

sqrt{9}=3

because

3*3=9

Actually there is one result more, which is -3, because:

(-3)*(-3)=9

A square root of a number can also be written as the number raised to the exponent ½:

sqrt{9}=9^{frac{1}{2}}

It is not possible to find the square root of a number smaller than zero, because you will not be able to find numbers that can be multiplied together to give a negative number, unless you are studying mathematics at a very high level and know about complex numbers.

Example: Cubic root

The cubic root or cube root of a number is the value that can be multiplied by itself three times to give the original number.

The cubic root is also denoted with the radical symbol, but with index 3 in front of it.

sqrt[3]{a}

For example, we want to find the cubic root of 27:

sqrt[3]{27}=3

because

3*3*3=27

There is only one result, when finding the cubic root, and in contrast to the square root it is possible to get a negative number as result.

A cubic root can be rewritten to the number raised to 1/3.


sqrt[3]{27}=27^{frac{1}{3}}

Finding roots in general

In general you can find two roots, if the index is an even number, like when you are finding the second, the fourth, the sixth root etc.:

sqrt[2]{a}, sqrt[4]{a}, sqrt[6]{a}

For roots with an odd number as index, there is only one root.

Properties of roots

Since roots can be rewritten into numbers with exponents, the propertyes for exponents also applies for roots:

sqrt[m]{a}*sqrt[n]{a}=sqrt[m*n]{a^{m+n}}

sqrt[m]{a}*sqrt[m]{b}=sqrt[m]{a*b}

frac{sqrt[m]{a}}{sqrt[n]{a}}=sqrt[m*n]{a^{n-m}}

frac{sqrt[m]{a}}{sqrt[m]{b}}=sqrt[m]{frac{a}{b}}

sqrt[n]{sqrt[m]{a}}=sqrt[m*n]{a}

sqrt[n]{a^m}=a^{frac{m}{n}}

(sqrt[n]{a})^m=a^{frac{m}{n}}