Tutorial: Using exponent identities
Adaptive game version
(This topic is also in Section 0.3 in Finite Mathematics and Applied Calculus) #[I don't like this new tutorial. Take me back to the older tutorial!][No me gusta este nueve tutorial. ¡Regresame al tutorial más viejo!]#
Positive exponent form, power form, and simplest radical form
In %%partAtut we saw that it is possible to write expressions with integral exponents in positive exponent form and in power form. At the end of %%partBtut we saw that it is also possible to write expressions with rational exponents in similar ways. For instance,
\t !r! $\dfrac{4}{7}x^{-5/2}$ \t ${}={}$ \t $\dfrac{4}{7x^{5/2}}$
\\ \t #[Power form][Forma potencia]# \t \t #[Positive exponent form][Forma exponente positiva]#
\\ \t \t \t #[(See %%partAtut)][(Ve %%partAtut)]#
\\
\\ \t !r! $\dfrac{4}{7x^{5/2}}$ \t ${}={}$ \t $\dfrac{4}{7\sqrt{x^5}}\ $ %%or $\ \dfrac{4}{7\left(\sqrt{x}\right)^5}$
\\ \t \t \t #[Radical form: Convert rational exponents to radicals.][Forma radical: Convertir exponentes racionales en radicales.]#
\\ \t \t \t #[(See %%partBtut)][(Ve %%partBtut)]#
\\
\\ \t !r! $\dfrac{4}{7\sqrt{x^5}}$ \t ${}={}$ \t $\dfrac{4}{7\sqrt{x^4\cdot x}} = \dfrac{4}{7x^2\sqrt{x}}$
\\ \t \t \t #[Take the $x^4$ out of the radical.][Saca el $x^4$ del radical.]#
\\ \t \t \t #[(See the second quiz in %%partBtut)][(Ve el segundo concurso en %%partBtut)]#
The last form shown on the right is called simplest radical form as the radical contains no factors that are squares (we took out the $x^4$ which is the square of $x^2$).
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 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} $
-
$\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 Term with one variable; $a$ = constant, $x$ = variable, $n$ = any power
\\ $ax^my^n$ $\qquad$ \t Term with two variables; $a$ = constant, $x, y$ = variables, $n, m$ = any powers
\\ $ax^my^nz^k$ $\qquad$ \t Term with three variables; $a$ = constant, $x, y, z$ = variables, $n, m, k$ = any powers
%%Examples 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}$
-
$\dfrac{3x}{4}\qquad $ $ \dfrac{3y^{-2}}{y}\qquad $ $ \dfrac{2}{4y^{-3}}\qquad $ $ y + \dfrac{1}{y}\qquad $ $ \dfrac{2}{3x^{-1}}$
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 of expressions in simplest radical form
Suggested video for this topic: Video by OrlandoHSMath
$\sqrt{3}\qquad$ $\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 #[The power under the radical can be made smaller.
Rewrite it as][La potencia bajo el radical puede hacerse más pequeña.
Reescribirlo como]# $\sqrt{2^2 \cdot 2} = 2\sqrt{2}$. \\ $\sqrt[3]{72} $ \t #[The power under the radical can be made smaller.
Rewrite it as][La potencia bajo el radical puede hacerse más pequeña.
Reescribirlo como]# $\sqrt[3]{8 \cdot 9} = \sqrt[3]{2^3 \cdot 9} = 2\sqrt[3]{9}$. \\ $5\sqrt{z^5} $ \t #[The power under the radical can be made smaller.
Rewrite it as][La potencia bajo el radical puede hacerse más pequeña.
Reescribirlo como]# $5\sqrt{z^4 \cdot z} = 5z^2\sqrt{z}$. \\ \\ $3\left(\sqrt{z}\right)^5 $ \t #[The power of the radical can be made smaller.
Rewrite it as][La potencia del radical puede hacerse más pequeña.
Reescribirlo como]# $3\left(\sqrt{z}\right)^4\sqrt{z} = 3z^2\sqrt{z}$. \\ \\ $\dfrac{2}{4x^{3/2}}$ \t #[Contains a fractional exponent.
Rewrite it as][Contiene un exponente no entero.
Reescribirlo como]# $\dfrac{2}{4\sqrt{x^3}} = \dfrac{2}{4x\sqrt{x}}$. \\ \\ $\sqrt{z^{-1}}$ \t #[Contains a negative exponent.
Rewrite it as][Contiene un exponente negativo.
Reescribirlo como]# $\dfrac{1}{\sqrt{z}}$ \\ \\ $3\left(\sqrt[7]{z}\right)^{10} $ \t #[The power of the radical can be made smaller.
Rewrite it as][La potencia del radical puede hacerse más pequeña.
Reescribirlo como]# $3\left(\sqrt[7]{z}\right)^{10}$ $= 3\left(\sqrt[7]{z}\right)^{7}\left(\sqrt[7]{z}\right)^{3}$ $= 3z\left(\sqrt[7]{z}\right)^{3}$.
%%Note #[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,]#
Rewrite it as][La potencia bajo el radical puede hacerse más pequeña.
Reescribirlo como]# $\sqrt{2^2 \cdot 2} = 2\sqrt{2}$. \\ $\sqrt[3]{72} $ \t #[The power under the radical can be made smaller.
Rewrite it as][La potencia bajo el radical puede hacerse más pequeña.
Reescribirlo como]# $\sqrt[3]{8 \cdot 9} = \sqrt[3]{2^3 \cdot 9} = 2\sqrt[3]{9}$. \\ $5\sqrt{z^5} $ \t #[The power under the radical can be made smaller.
Rewrite it as][La potencia bajo el radical puede hacerse más pequeña.
Reescribirlo como]# $5\sqrt{z^4 \cdot z} = 5z^2\sqrt{z}$. \\ \\ $3\left(\sqrt{z}\right)^5 $ \t #[The power of the radical can be made smaller.
Rewrite it as][La potencia del radical puede hacerse más pequeña.
Reescribirlo como]# $3\left(\sqrt{z}\right)^4\sqrt{z} = 3z^2\sqrt{z}$. \\ \\ $\dfrac{2}{4x^{3/2}}$ \t #[Contains a fractional exponent.
Rewrite it as][Contiene un exponente no entero.
Reescribirlo como]# $\dfrac{2}{4\sqrt{x^3}} = \dfrac{2}{4x\sqrt{x}}$. \\ \\ $\sqrt{z^{-1}}$ \t #[Contains a negative exponent.
Rewrite it as][Contiene un exponente negativo.
Reescribirlo como]# $\dfrac{1}{\sqrt{z}}$ \\ \\ $3\left(\sqrt[7]{z}\right)^{10} $ \t #[The power of the radical can be made smaller.
Rewrite it as][La potencia del radical puede hacerse más pequeña.
Reescribirlo como]# $3\left(\sqrt[7]{z}\right)^{10}$ $= 3\left(\sqrt[7]{z}\right)^{7}\left(\sqrt[7]{z}\right)^{3}$ $= 3z\left(\sqrt[7]{z}\right)^{3}$.
- $\sqrt{a^2b} = |a|\sqrt{b}$
Suggested video for this topic: Video by OrlandoHSMath
Converting between positive exponent, power, and simplest radical forms
We can use the exponent identities to convert expressions from one form to the other. The table below shows some examples, followed by some for you to do yourself.
- To enter $\sqrt{A}$ type sqrt(A).
To enter][Como ingresar radicales:
- Para ingresar $\sqrt{A}$ tecla sqrt(A).
Para ingresar ]# $\sqrt[n]{A}$ #[ type ][ tecla ]# sqrt[n](A).
Now try the exercises in Section 0.3 in Finite Mathematics and Applied Calculus.
or move ahead to the next tutorial by pressing "Next tutorial" on the sidebar.
Copyright © 2019 Stefan Waner and Steven R. Costenoble