Raional exponents
#[If $a$ is a real number
other than zero and $n$ is a positive integer, then we define.][Si $a$ es un número real
distinto de cero y $n$ es un número entero positivo, entonces definimos.]#
We can use rational exponents for expressions involving radicals as follows:
#[Radical form][Forma radical]# \t \t #[Exponent form][Forma exponente]# \t \t #[Example][Ejemplo]#
\\ $\sqrt{a}$ \t $\quad$ \t $a^{1/2} \quad (\text{or } a^{0.5})$ \t $\quad$ \t $64^{1/2} = \sqrt{64} = 8$
\\ $\sqrt[3]{a}$ \t $\quad$ \t $a^{1/3}$ \t $\quad$ \t $64^{1/3} = \sqrt[3]{64} = 4$
\\ $\sqrt[n]{a}$ \t $\quad$ \t $a^{1/n}$ \t $\quad$ \t $64^{1/6} = \sqrt[6]{64} = 2$
So, if we want the exponent identities to work with rational exponents, we can calculate $a^{m/n}$ in two ways:
\t $a^{m/n} = a^{(m)(1/n)} = (a^m)^{1/n} = \sqrt[n]{a^m}$\t $\qquad$ \t
\\ #[or][o]# $\quad$
\\ \t $a^{m/n} = a^{(1/n)(m)} = (a^{1/n})^{m} = \left(\sqrt[n]{a}\right)^m.$
#[Examples][Ejemplos]#
$4^{1/2} = \sqrt{4} = 2$
\\ $4^{3/2} = (4^3)^{1/2} = 64^{1/2} = 8$
\\ $4^{3/2} = (4^{1/2})^3 = \left(\sqrt{4}\right)^3 = 2^3 = 8$
#[Some for you][Algunos para ti]#