Radical Functions

Roots are the inverse operation for exponents. An expression with roots is called a radical expression. It's easy, although perhaps tedious, to compute exponents given a root. For instance $7\cdot 7\cdot 7\cdot 7 = 49\cdot 49 = 2401$.

If fourth root of 2401 is 7, and the square root of 2401 is 49, then what is the third root of 2401?

Finding the value for a particular root is difficult. This is because exponentiation is a different kind of function than addition, subtraction, multiplication, and division. When graphing functions, expressions that use exponentiation use curves instead of lines. Using algebra will show that not all of these expressions are functions and that knowing when an expression is a relation or a function allows certain types of assumptions to be made. These assumptions can be used to build mental models for topics that would otherwise be impossible to understand.

For now, deal with roots by turning them back into exponents. If a root is defined as the $n$th root of $x$, it is represented as $\sqrt [ n ]{ x } = r$ . Get rid of the root by raising the answer to the nth power, i.e. ${r}^{n}=x$

Square root

If the square root of a number is taken, the result is a number which when squared gives the first number. This can be written symbolically as:$\sqrt x = y$ if ${y}^{2}=x$.

In 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.

Cube roots

Roots do not have to be square. The cube root of a number ($\sqrt [ 3 ]{x}$ ) can also be taken. The cube root is the number which, when cubed, or multiplied by itself and then multiplied by itself again, gives back the original number. For example, the cube root of 8 is 2 because $2 \cdot 2\cdot 2=8$ , or $\sqrt[3]{8}=2$

Other roots

There are an infinite number of possible roots all in the form of $\sqrt [n]{a}$ which corresponds to ${a}^{\frac{1}{n}}$, when expressed using exponents. If $\sqrt [n]{a} = b$ then ${b}^{n} = a$ . The only exception is 0. $\sqrt [0]{a}$ is undefined, as it corresponds to ${a}^{\frac{1}{0}}$, resulting in a division by zero. Even if attempting to discover the 0th root of 1, no progress will be made, as practically any number to the power of zero equals 1, leaving only an undefined result.

Graphs of Radical Functions

Since roots are simply the inverse of exponents, graphing roots can be seen as just graphing exponents with the axes reversed. The shape of the radical graph will resemble the shape of the related exponent graph it were rotated 90-degrees clockwise. For example, the graph of $y=\sqrt x$ is the graph of $y=x^2$rotated clockwise. Note that half of the parabola is missing since functions cannot have more than one value at a point, and the square root function is taken to yield a positive value (though $(-x)^2$ gives the same value as $x^2$ so the square root of a number $y$ such that $y=x^2$ would be $\sqrt y = \pm x$). It is important to remember when graphing the roots that negative values of $x$ will not produce real numbers. This will be explained further in the section on imaginary numbers.

Irrational numbers

If a root of a whole number is squared root, which is not itself the square of a rational number, the answer will have an infinite number of decimal places. Such a number is described as irrational and is defined as a number which cannot be written as a rational number: $\displaystyle \frac {a}{b}$, where $a$ and $b$ are integers. However, using a calculator can approximate the square root of a non-square number:$\sqrt {3} = 1.73205080757$

The result of taking the square root is written with the approximately equal sign because the result is an irrational value which cannot be written in decimal notation exactly. Writing the square root of 3 or any other non-square number as $\sqrt {3}$ is the simplest way to represent the exact value. Irrational numbers also appear when attempting to take cube roots or other roots. However, they are not restricted to roots, and may also appear in other mathematical constants (e.g. π, e, φ, etc.).