Math & History: Roots (Radicals)

I. Intro

A radical expression is any mathematical expression containing a radical symbol (√).
Radicals are simply the inverse operation to exponents. More technically, the root of a number A is another number x, which when multiplied by itself n number of times, equals A.

Hence, ${\sqrt[{n}]{A}}$ where $x^{n}=A$

There are three parts of a radical expression. The radical symbol, which means "root of." The radicand, which is the number under the radical symbol, is the number you are finding the "root of." The index, which is the small number written as part of the radical symbol, indicates the number of times the unknown number is multiplied by itself to equal the radicand. If no index is shown, it is assumed to be 2, or square root.

Roots
The 2nd root is generally called the square root.
The 3rd root is called the cube root.
Roots higher than three are referred by ordinary numbers such as "fourth root", "fifth root", etc.

Verbalizing radicals involves reading the index (root) followed by the radicand (expression inside):

Roots can also be written in exponential form, such as:

II. Square Roots
When we are trying to find the square root of a number (say, 25), we are trying to find a number that when multiplied by itself gives that original number. In the case of 25, we find that 5⋅5=25, so 5 is the square root of 25.

The square root is the inverse of the square (exponent of 2), much like multiplication is the inverse of division. A perfect square is any whole number that has been squared. Consequently, all perfect squares have square roots that are whole numbers.

\sqrt{25} = 5

Here are some other simple examples:

Principal Root

The square root of a number is the value that, when multiplied by itself, gives the original number. For example, both +5 and −5 are square roots of 25 because:

5×5=25

(−5)×(−5)=25

Thus, every positive real number n has two square roots—a positive square root and a negative square root. The unique positive square root is called the principal square root or principal root.

The radical sign (√) generally represents the principal square root. Unless specified otherwise, “the square root” of a number refers exclusively to the principal square root.

Connection To A Square

Square Root of a Decimal

Khan: Square Root of Decimal

Square Root Decimal by Converting to Fraction

Square Root of a Fraction

Onlinemath: Simplify Square Roots That Have Fractions

III. Cube Root

When we are trying to find the cube root of a number (say, 27), we are trying to find a number that when multiplied by itself three times gives that original number. In the case of 27, we find that 3⋅3⋅3=27, so 3 is the cube root of 27.

Factoring to find Cube Root

If we can't figure out what factor multiplied by itself three times will result in the given number, we can make a factor tree.

Example: