Simplest radical form
An algebraic expression or a number is in
simplest radical form if it is written using radicals and only positive integer exponents, powers under the radical are as small as possible, and powers of radicals are as small as possible.
*
* Some people also insist that there can be no radicals in the denominator. We see no good reason for this restiction; in fact, removing radicals from the denominator frequently results in an expression that is less simple.
#[Examples][Ejemplos]# of expressions in simplest radical form
$\sqrt{3}\qquad$ $\dfrac{\sqrt{3}}{\sqrt{2}}\qquad $ $\dfrac{3z\sqrt{z}}{4y^5}\qquad $ $ \dfrac{2}{4\sqrt[3]{x^2}}$ $ \qquad $ $3\left(\sqrt[9]{z}\right)^8 $
#[The following expressions are
not in simplest radical form:][Las siguientes expresiones
no son de la forma radical más simple:]#
$\sqrt{8} $ \t $\sqrt{2^2 \cdot 2} = 2\sqrt{2}$.
\\ $\sqrt[3]{72} $ \t $\sqrt[3]{8 \cdot 9} = \sqrt[3]{2^3 \cdot 9} = 2\sqrt[3]{9}$.
\\ $5\sqrt{z^5} $ \t $5\sqrt{z^4 \cdot z} = 5z^2\sqrt{z}$.
\\
\\ $3\left(\sqrt{z}\right)^5 $ \t $3\left(\sqrt{z}\right)^4\sqrt{z} = 3z^2\sqrt{z}$.
\\
\\ $\dfrac{2}{4x^{3/2}}$ \t $\dfrac{2}{4\sqrt{x^3}} = \dfrac{2}{4x\sqrt{x}}$.
\\
\\ $\sqrt{z^{-1}}$ \t $\dfrac{1}{\sqrt{z}}$
\\
\\ $3\left(\sqrt[7]{z}\right)^{10} $ \t $3\left(\sqrt[7]{z}\right)^{10}$ $= 3\left(\sqrt[7]{z}\right)^{7}\left(\sqrt[7]{z}\right)^{3}$ $= 3z\left(\sqrt[7]{z}\right)^{3}$.