In a previous article we went through integer exponents. In this tutorial we are going to learn about a bit more complicated topic, rational exponents.
A rational exponent has the form:
\[ \begin{equation} \begin{split}\bbox[#FFFF9D]{x^{\frac{m}{n}}}
\end{split} \end{equation} \]
where both m and n are integer numbers.
As you can see, when we are dealing with rational exponent we basically use some numbers raised at the power of a fraction. In order to be rational, the exponent is a fraction of two integer numbers.
A particular case of rational exponents is:
\[x^{\frac{1}{n}}\]where n is an integer number.
This expression is called the n(th) root of x. Example:
\[ \begin{equation*} \begin{split}x^{\frac{1}{2}}{\hspace 5mm}&is{\hspace 2mm}square{\hspace 2mm}root{\hspace 2mm}of{\hspace 2mm}x\\
x^{\frac{1}{3}}{\hspace 5mm}&is{\hspace 2mm}cubic{\hspace 2mm}root{\hspace 2mm}of{\hspace 2mm}x
\end{split} \end{equation*} \]
Let’s define a number y as:
\[ \begin{equation} \begin{split}\bbox[#FFFF9D]{y=x^{\frac{1}{n}}}
\end{split} \end{equation} \]
If we raise y at the power of n, we get:
\[y^n=\left (x^{\frac{1}{n}} \right )^n = x^{\frac{1}{n} \cdot n}=x\]So, when solving a exponentiation problem of the form (1) we need to find the number x raised at the power n which gives y.
Examples:
\[ \begin{equation*} \begin{split}16^{\frac{1}{4}}=2 {\hspace 5mm}&because{\hspace 5mm} 2^4 = 16\\
49^{\frac{1}{2}}=7 {\hspace 5mm}&because{\hspace 5mm} 7^2 = 49\\
-27^{\frac{1}{3}}=-3 {\hspace 5mm}&because{\hspace 5mm} \left (-3 \right )^3 = -27
\end{split} \end{equation*} \]
The proprieties of integer exponents are valid also for rational exponents. If you want your memory refreshed, have a look at the tutorial Exponentiation – integer exponents.
With the properties in mind we are going through a simplification type example. Simplify the mathematical expression below, writing the answer only with positive exponents.
\[ \begin{equation*} \begin{split}\left (\frac{x^{-4}}{25 \cdot y^{\frac{1}{3}}} \right )^{\frac{1}{2}}
\end{split} \end{equation*} \]
Solution:
\[ \begin{equation*} \begin{split}\left (\frac{x^{-4}}{25 \cdot y^{\frac{1}{3}}} \right )^{\frac{1}{2}}=\frac{\left ( x^{-4} \right )^{\frac{1}{2}}}{\left ( 25 \cdot y^{\frac{1}{3}} \right )^{\frac{1}{2}}}=\frac{x^{-4 \cdot \frac{1}{2}}}{25^{\frac{1}{2}}\cdot y^{\frac{1}{3} \cdot \frac{1}{2}}}=\frac{x^{-2}}{5 \cdot y^{\frac{1}{6}}}=\frac{1}{5 \cdot x^{2} \cdot y^{\frac{1}{6}}}
\end{split} \end{equation*} \]
As you can see, the rational exponents problems are handled in the same manner as integer exponents. If the properties of integer exponents are well known, simplification type problems will be easy to handle.
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