Basic Technical Mathematics, 11th Edition

11.2 Fractional Exponents

• Meaning of Fractional Exponents • Evaluations • Graphs • Simplifying Expressions

In Section 11.1, we reviewed the use of integer exponents, including exponents that are negative integers and zero. We now show how rational numbers may be used as exponents. With the appropriate definitions, all the laws of exponents are valid for all rational numbers as exponents.

Equation (11.3) states that ${(a^m)}^n\hspace{0.17em}=\hspace{0.17em}{a}^{mn}\hspace{0.17em}.\hspace{0.17em}$ If we were to let $m\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{1}{2}}$ and $n\hspace{0.17em}=\hspace{0.17em}2\hspace{0.17em},\hspace{0.17em}$ we would have ${(a^{\mathrm{1/2}})}^2\hspace{0.17em}=\hspace{0.17em}{a}^1\hspace{0.17em}.\hspace{0.17em}$ However, we already have a way of writing a quantity that, when squared, equals a. This is written as $\sqrt{a}\hspace{0.17em}.\hspace{0.17em}$ To be consistent with previous definitions and to allow the laws of exponents to hold, we define

$$a^{1\hspace{0.17em}/\hspace{0.17em}n}\hspace{0.17em}=\hspace{0.17em}\sqrt[n]{a}$$(11.7)

The radical in Eq. (11.7 ...