Basic Technical Mathematics, 11th Edition

11.1 Simplifying Expressions with Integer Exponents

• Laws of Exponents • Simplifying Expressions • Zero and Negative Exponents

The laws of exponents were given in Section 1.4. For reference, they are

$$a^m\hspace{0.17em}\times \hspace{0.17em}{a}^n\hspace{0.17em}=\hspace{0.17em}{a}^{m\hspace{0.17em}+\hspace{0.17em}n}$$(11.1)

$${\displaystyle \frac{a^m}{a^n}}\hspace{0.17em}=\hspace{0.17em}{a}^{m\hspace{0.17em}-\hspace{0.17em}n}\text{or }{\displaystyle \frac{a^m}{a^n}}\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{1}{a^{n\hspace{0.17em}-\hspace{0.17em}m}}}(a\hspace{0.17em}\ne \hspace{0.17em}0)$$(11.2)

$${(a^m)}^n\hspace{0.17em}=\hspace{0.17em}{a}^{mn}$$(11.3)

$${(ab)}^n\hspace{0.17em}=\hspace{0.17em}{a}^n{b}^n\hspace{0.17em},\hspace{0.17em}{\left({\displaystyle \frac{a}{b}}\right)}^n\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{a^n}{b^n}}(b\hspace{0.17em}\ne \hspace{0.17em}0)$$(11.4)

$$a^0\hspace{0.17em}=\hspace{0.17em}1(a\hspace{0.17em}\ne \hspace{0.17em}0)$$(11.5)

$$a^{\hspace{0.17em}-\hspace{0.17em}n}\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{1}{a^n}}(a\hspace{0.17em}\ne \hspace{0.17em}0)$$(11.6)

Although Eqs. (11.1) to (11.4) were originally defined for positive integers as exponents, we showed in Section 1.4 that with the definitions in Eqs. (11.5) and (11.6), they are valid for all integer exponents. Later in the chapter, we will show how fractions are used as exponents. These equations are very important to the developments of this chapter and should be reviewed and ...