Roots are the inverse operation of exponentiation. Mathematical expressions with roots are called radical expressions and can be easily recognized because they contain a radical symbol ($\sqrt{}$).

Recall that exponents signify that we should multiply a given integer a certain number of times. For example, $7^2$ tells us that we should multiply 7 by itself two times:

$7^2 = 7 \cdot 7 = 49$

Since roots are the inverse operation of exponentiation, they allow us to work backwards from the solution of an exponential expression to the number in the base of the expression. 

For example, the following is a radical expression that reverses the above solution, working backwards from 49 to its square root: 

$\sqrt{49} = 7$

In this expression, the symbol is known as the "radical," and the solution of 7 is called the "root." 

Finding the value for a particular root can be much more difficult than solving an exponential expression. For now, it is important simplify to recognize the relationship between roots and exponents: if a root $r$ is defined as the $n \text{th}$ root of $x$, it is represented as 

$\sqrt [ n ]{ x } = r$

Because roots are the inverse of exponents, we can cancel out the root in this equation by raising the answer to the nth power: 

$\left( \sqrt [ n ]{ x }\right) ^n = \left(r\right) ^n$

To simplify:

${r}^{n}=x$

Square Roots

If the square root of a number $x$ is calculated, the result is a number that when squared (i.e., when raised to an exponent of 2) gives the original number $x$. This can be written symbolically as follows: $\sqrt x = y$ if ${y}^{2}=x$. This rule applies to the series of real numbers ${ y }^{ 2 }\ge 0$, regardless of the value of $y$. As such, when $x<0$ then $\sqrt x$ cannot be defined.

For example, consider the following: $\sqrt{36}$. This is read as "the square root of 36" or "radical 36." You may recognize that $6^2 = 6 \cdot 6 = 36$, and therefore conclude that 6 is the root of $\sqrt{36}$. Thus we have the answer, $\sqrt{36} = 6$.

Cube Roots

The cube root of a number ($\sqrt [ 3 ]{x}$ ) can also be calculated. The cube root of a value $x$ is the number that when cubed (i.e., when raised to an exponent of 3) yields the original number $x$. 

For example, the cube root of 8 is 2 because $2^3 = 2 \cdot 2\cdot 2=8$. This can also be written as $\sqrt[3]{8}=2$.

Other Roots

There are an infinite number of possible roots all in the form of $\sqrt [n]{a}$ . Any non-zero integer can be substituted for $n$. For example, $\sqrt[4]{a}$ is called the "fourth root of $a$," and $\sqrt[20]{a}$ is called the "twentieth root of $a$." 

Note that for any such root, if $\sqrt [n]{a} = b$ then ${b}^{n} = a$. As an example, consider $\sqrt[4]{2401} = 7$ . $7^4 = 7\cdot 7\cdot 7\cdot 7 = 2401$.