Radicals

In this tutorial we are going to learn about radicals. In mathematics a radical is described as:

\[\begin{equation} \begin{split} \bbox[#FFFF9D]{\sqrt[n]{x}} \end{split} \end{equation}\]

where:

  • n is a positive integer, greater than 1
  • x is a real number

The number n is called the order/index of the radical, the number x is called the radicand, and the symbol  is called the radical.

The radical can be written as a rational exponent as follows:

\[\begin{equation} \begin{split} \bbox[#FFFF9D]{\sqrt[n]{x}=x^{\frac{1}{n}}} \end{split} \end{equation}\]

There are two special cases of radicals:

  • square root, when the order of the radical is 2:
\[\begin{equation} \begin{split} \sqrt{x} \end{split} \end{equation}\]
  • cubic root, when the order of the radical is 3:
\[\begin{equation} \begin{split} \sqrt[3]{x} \end{split} \end{equation}\]

There a two things to keep in mind about the square root:

  • the index of the radical is not written down, this mean that every radical without the index is a square root
  • the radicand x is always a positive number, because there is no real number raised at the power of two which can result in a negative number (except when it is a complex number)

In the table below we can see the main properties of radicals together with some examples. Bear in mind that, since a radical can be written as a exponent, all exponents (integer or rational) properties are valid also for radicals.

#PropertyExamples
1\[ \begin{equation*} \begin{split}
\sqrt[n]{x^n} = x
\end{split} \end{equation*} \]
\[ \begin{equation*} \begin{split}
\sqrt{25} = \sqrt{5^2} = 5\\
\sqrt[3]{27} = \sqrt[3]{3^3} = 3\\
\sqrt[4]{16} = \sqrt[4]{2^4} = 2
\end{split} \end{equation*} \]
2\[ \begin{equation*} \begin{split}
\sqrt[n]{x \cdot y} = \sqrt[n]{x} \cdot \sqrt[n]{y}
\end{split} \end{equation*} \]
\[ \begin{equation*} \begin{split}
\sqrt{9 \cdot 4} =& \sqrt{36} = 6\\
\sqrt{9 \cdot 4} =& \sqrt{9} \cdot \sqrt{4} = 3 \cdot 2 = 6
\end{split} \end{equation*} \]
3\[ \begin{equation*} \begin{split}
\sqrt[n]{\frac{x}{y}} = \frac{\sqrt[n]{x}}{\sqrt[n]{y}}
\end{split} \end{equation*} \]
\[ \begin{equation*} \begin{split}
\sqrt{\frac{36}{4}} =& \sqrt{9} = 3\\
\sqrt{\frac{36}{4}} =& \frac{\sqrt{36}}{\sqrt{4}} = \frac{6}{2} = 3
\end{split} \end{equation*} \]

The radical function is basically a rational exponent. A radical can be easily be written as a rational exponent and vice-versa. Example:

\[ \begin{equation*} \begin{split}
x^{\frac{m}{n}}=\left (x^{\frac{1}{n}} \right)^m = \left (\sqrt[n]{x} \right)^m
\end{split} \end{equation*} \]

which is the same as:

\[ \begin{equation*} \begin{split}
x^{\frac{m}{n}}=\left (x^{m} \right)^{\frac{1}{n}} = \sqrt[n]{x^m}
\end{split} \end{equation*} \]

Remember that the radical of the sum of two numbers is not equal to the sum of the radical of each number. The same rule apply for the difference:

\[ \begin{equation*} \begin{split}
\sqrt[n]{x+y} &\neq \sqrt[n]{x} + \sqrt[n]{y}\\
\sqrt[n]{x-y} &\neq \sqrt[n]{x} – \sqrt[n]{y}
\end{split} \end{equation*} \]

Example:

\[5=\sqrt{25}=\sqrt{9+16}\neq\sqrt{9}+\sqrt{16}=3+4=7\]

With the properties in mind we are going to solve a simplification problem. Simplify the mathematical expressions below assuming that x and y are positive numbers:

\[\sqrt[4]{16 \cdot x^8 \cdot y^2}\]

Solution:

\[ \begin{equation*} \begin{split}
\sqrt[4]{16 \cdot x^8 \cdot y^2} &= \sqrt[4]{16} \cdot \sqrt[4]{x^8} \cdot \sqrt[4]{y^2}\\
&=2^{\frac{4}{4}} \cdot x^{\frac{8}{4}} \cdot y^{\frac{2}{4}}\\
&=2 \cdot x^{2} \cdot y^{\frac{1}{2}}\\
&=2 \cdot x^{2} \cdot \sqrt{y}
\end{split} \end{equation*} \]

For reference, in order to have a quick access resource on radicals properties and examples, download the image below:

Radicals Properties and Examples

Image: Radicals Properties and Examples

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