Positive exponent form
An algebraic expression in
positive exponent form is one in which there are no radicals, and all exponents are positive.
Expressions in positive exponent form are often written using fractions with powers of variables in the numerator and/or the denominator.
#[Examples][Ejemplos]# of expressions in positive exponent form
$\dfrac{3z^2}{4y^5}\qquad $ $ \dfrac{2}{4x^3}\qquad $ $ 3.5z^8\qquad $ $ \dfrac{1}{x}\qquad $ $ \dfrac{x^{1/2}}{y^{2/3}}\qquad $ $ \dfrac{3}{4} \qquad $ $ \dfrac{2}{x} - \dfrac{4x^3}{z} $
The following expressions are
not in positive exponent form because they contain negative or zero exponents:
$\dfrac{3y^{-2}}{4y^5} \qquad $ $ \dfrac{2}{4x^{-3}} \qquad $ $ x^{-1} \qquad$ $3x^0 \qquad $ $x^{-1/2} \qquad$ $3^{-2} $
Power form
An algebraic expression in
power form is one in which there are no radicals, and none of the variables appear as part of a fraction (although the constants can be fractions).
Expressions in power form are typically written as sums and differences of terms of the following form:
$ax^n$ $\qquad$ \t
\\ $ax^my^n$ $\qquad$ \t
\\ $ax^my^nz^k$ $\qquad$ \t
#[Examples][Ejemplos]# of expressions in power form
$4z^{-2}\qquad $ $ \dfrac{2}{3}x^{-1/2}\qquad $ $ 3 + x - x^2\qquad $ $ 3x^2y^{-2}\qquad $ $ 4z^{-2} - 2y^{1/2}$
The following expressions are
not in power form because they contain variables that appear in fractions:
$\dfrac{3x}{4}\qquad $ $ \dfrac{3y^{-2}}{y}\qquad $ $ \dfrac{2}{4y^{-3}}\qquad $ $ y + \dfrac{1}{y}\qquad $ $ \dfrac{2}{3x^{-1}}$
For practice converting to power form, see %%partAtut.
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 $\sqrt{2^2 \cdot 2} = 2\sqrt{2}$.
\\
\\ $5\sqrt{z^5} $ \t $5\sqrt{z^4 \cdot z} = 5z^2\sqrt{z}$.
\\
\\ $\dfrac{2}{4x^{3/2}}$ \t $\dfrac{2}{4\sqrt{x^3}} = \dfrac{2}{4x\sqrt{x}}$.
\\
\\ $\sqrt{z^{-1}}$ \t $\dfrac{1}{\sqrt{z}}$
#[Note][Nota]# #[Be careful with square roots (or other even roots) of powers of letter variables; for instance,][Ten cuiodado con raiceas cuadradas (u otras raices pares) de potencias de variables letras, for ejemplo,]#
$\sqrt{a^2b} = |a|\sqrt{b}$
#[because $a$ may be negative. In the quizzes here, we will assume that all letter variables are positive to avoid that issue.][porque $a$ puede ser negativo. En los concursos aquí, asumiremos que todas las variables de letras son positivas para evitar esa situación.]#