Exponential Functions
We will go over properties of exponential functions.
The Essentials
Two exponents with the same term can be multiplied or divided together:
\[
a^m a^n = a^{m+n}
\]
(1)
\[
\frac{a^m}{a^n} = a^{m-n}
\]
(2)
An exponential expression can be raised to a power:
\[
(a^m)^n = a^{mn}
\]
(3)
In a product or quotient of two expressions raised to a power, the exponent can be distributed:
\[
(ab)^m = a^m b^m
\]
(4)
\[
\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}
\]
(5)
A negative exponent means inverting the equivalent expression with a positive exponent:
\[
a^{-m} = \frac{1}{a^m}
\]
(6)
A rational exponent means a radical term:
\[
a^{1/m} = \sqrt[m]{a}
\]
(7)
\[
a^{{n}/{m}} = \sqrt[m]{a^n}
\]
(8)
Example
Evaluate the expression: \( \left(\frac{4}{9}\right)^{3/2}\)
\[
\left(\frac{4}{9}\right)^{3/2}
\]
(using(3))
\[
\left( \left(\frac{4}{9}\right)^{1/2}\right)^3
\]
\[
\left( \frac{4^{1/2}}{9^{1/2}} \right)^3
\]
(using(5))
\[
\left( \frac{ \sqrt{4} } { \sqrt{9} } \right)^3
\]
(using(7))
\[
\left( \frac{2}{3} \right)^3
\]
\[
\frac{2^3}{3^3}
\]
(using(5))
\[
\frac{8}{27}
\]
Practice
Evaluate these expressions:
- \( 125^{4/3} \)
- \( \left(\frac{16}{9}\right)^{-3/2} \)
- \( x^{4/3} \cdot x^{3/2} \)
Solution:
- 625
- \( \frac{27}{64} \)
- \( x^{17/6} \)