Tutorial: Exponents and radicals
Adaptive game version
This tutorial: Part A: Integer exponents
(This topic is also in Section 0.2 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 integer exponents
"Raising a number to a positive integer power" means multiplying that number by tself that number of times. In general, an exponent refers to a power to which a number can be raised. In Part A of this tutorial we focus on integer exponents, starting with positive integer exponents.
Positive integer exponents
If $a$ is a real number and $n$ is a positive integer, then by $a^n$ we mean the quantity $a \cdot a\cdot a\ \cdots \ a \qquad$ ($n$ times). %%Therefore,
$a^1 = a$\\
$a^2 = a \cdot a$\\
$a^3 = a \cdot a \cdot a$\\
$a^4 = a \cdot a \cdot a \cdot a$\\
#[and so on.][y así sucesivamente.]# #[The number $a$ is called the base and the number $n$ is called the exponent.][El número $a$ se llama la base y el número $n$ se llama el exponente.]#
Suggested video for this topic: Video by eHowEducation
Examples
$3^2 = 3 \cdot 3 = 9$ \t Base $3,$ exponent $2$ \\
$2^3 = 2 \cdot 2 \cdot 2 = 8$ \t Base $2,$ exponent $3$\\
$0^{34} = 0 \cdot 0 \ \cdots \ 0 = 0$ \t $0$ to any positive power is $0.$ \\
$1^{34} = 1 \cdot 1 \ \cdots \ 1 = 1$ \t $1$ to any positive power is $1.$ \\
$(-1)^{3} = (-1) \cdot (-1) \cdot (-1) = -1$
Caution! The meaning of −xn
A negative sign at the start of an expression indicates multiplication by $-1$. So, $-x^n$ means $(-1)x^n$ and is calculated using the standard order of operations: First take the power, and then multiply by $-1$. For instance,
Spreadsheets and some programming languages will interpret $-3^2$ (wrongly!) as $(-3)^2 = 9,$ so be on the lookout for that when working with your spreadsheet, where you should write it as $(-1)3^2$ to prevent that from happening.
$-3^2$ \t $= (-1)3^2$ \tab \t Meaning of negative sign at start of expression
\\ \t $= (-1)9$ \tab \t Calculate the power first.
\\ \t $= -9.$ \tab \t Multiply by −1.
Spreadsheets and some programming languages will interpret $-3^2$ (wrongly!) as $(-3)^2 = 9,$ so be on the lookout for that when working with your spreadsheet, where you should write it as $(-1)3^2$ to prevent that from happening.
Some for you
Negative and zero exponents
In mathematics we also give meaning to the notion of "raising a number to a negative power":
Negative and zero exponents
If $a$ is a real number other than zero and $n$ is a positive integer, then we define
#[A][R]#: Consider the following example: Start with, say, $2^4$ and then successively decrease the exponent by 1 each time:
$a^0 = 1$ \t The zeroth power of $a$ is $1.$ \\
$a^{-n} = \dfrac{1}{a^n} = \dfrac{1}{a \cdot a\cdot a\ \cdots \ a} \qquad $ ($n$ #[times][veces]#).
#[Therefore][Por lo tanto]#,
$a^{-1} = \dfrac{1}{a}$\\
$a^{-2} = \dfrac{1}{a^2} = \dfrac{1}{a \cdot a}$\\
$a^{-3} = \dfrac{1}{a^3} = \dfrac{1}{a \cdot a \cdot a}$\\
$a^{-4} = \dfrac{1}{a^4} = \dfrac{1}{a \cdot a \cdot a \cdot a}$\\
and so on.
#[Q][P]#: Where do these rules come from? #[A][R]#: Consider the following example: Start with, say, $2^4$ and then successively decrease the exponent by 1 each time:
\t !2! \gap[40] $2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16$
\\ \t \gap[40] $2^3 = 2 \cdot 2 \cdot 2 = 8$ \t Divide the preceding answer, $16$, by $2$. \t \gap[80]
\\ \t \gap[40] $2^2 = 2 \cdot 2 = 4$ \t Divide the preceding answer, $8$, by $2$.
\\ \t \gap[40] $2^1 = 2$ \t Divide the preceding answer, $4$, by $2$.
\\
\\ \t !3! #[We see that each answer can be obtained from the answer above by dividing by 2. In other words, decreasing the exponent by 1 is the same as dividing by the base 2. So, if we continue this process, the next line ought to be][Vemos que cada resultado se puede obtener del resultado anterior dividiendo por 2. En otras palabras, disminuir el exponente en 1 es lo mismo que dividir por la base 2. Entonces, si continuamos con este proceso, la siguiente línea debería ser]#
\\
\\ \t \gap[40] $2^0 = 1$ \t Divide the preceding answer, $2$, by $2$.
\\ \t \gap[40] $2^{-1} = \dfrac{1}{2}$ \t Divide the preceding answer, $1$, by $2$.
\\ \t \gap[40] $2^{-2} = \dfrac{1}{2^2} = \dfrac{1}{4}$ \t Divide the preceding answer, $\dfrac{1}{2}$, by $2$.
\\ \t \gap[40] ...
Suggested video for this topic: Video by BespokeEducation
Examples
$3^0 = 1; \qquad 654^0 = 1; \qquad \pi^0 = 1$\\
$3^{-1} = \dfrac{1}{3}$\\
$3^{-2} = \dfrac{1}{3^2} = \dfrac{1}{9}$ \\
$2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$ \\
$1^{-34} = \dfrac{1}{1^{34}} = \dfrac{1}{1} = 1$ \\
$(-1)^{-3} = \dfrac{1}{(-1)^3} = \dfrac{1}{-1} = -1$ \\
$2^33^{-2} = 2^3 \cdot \dfrac{1}{3^2} = \dfrac{2^3}{3^2} = \dfrac{8}{9}$
Some for you
Laws of exponents
#[What happens if you multiply, say, $2^5$ and $2^6$? What happens if you divide one by the other, or raise the first one to the third power? The laws of exponents tell us what would happen in these and similar cases. For convenience, we have grouped the laws into two categories: those that combine exponential expressions with the same base, and those that combine expressions with different bases:][¿Qué sucede si multiplicas, por ejemplo, $2^5$ y $2^6$? ¿Qué sucede si divides una por la otra, o elevas la primera a la tercera potencia? Las leyes de los exponentes nos dicen lo que suceder�a en estos casos y en otros similares. Para mayor comodidad, hemos agrupado las leyes en dos categor�as: las que combinan expresiones exponenciales con la misma base y las que combinan expresiones con diferentes bases.]#
Suggested video for this topic: Video by Mometrix Academy
Exponent laws to combine expressions with the same base
| #[Law][Ley]# | %%Examples | #[Comments][Comentarios]# | |||||||||||
| 1. $a^ma^n = a^{m+n}$ |
$2^32^2 = 2^{3+2} = 2^5 = 32$
$2^32^{-2} = 2^{3-2} = 2^1 = 2$ $(-9)^4(-9)^{-2} = (-9)^{4-2} = (-9)^2 = 81$ $x^3x^4 = x^{3+4} = x^7$ $x^{-4}x^3 = x^{-4+3} = x^{-1} = \dfrac{1}{x}$ |
If the bases in a product match, add the exponents. If the bases do not match the rule does not apply. To see why this rule holds in the case of positive exponents, notice that
$\qquad \quad a^m+a^n \neq (a+a)^n$ |
|||||||||||
|
2. $\dfrac{a^m}{a^n} = a^{m-n}$ (%%if $a \neq 0$) |
$\dfrac{2^3}{2^2} = 2^{3-2} = 2^1 = 2$
$\dfrac{x^3}{x^4} = x^{3-4} = x^{-1} = \dfrac{1}{x}$ $\dfrac{2^3}{2^{-2}} = 2^{3-(-2)} = 2^5 = 32$ $\dfrac{x^{-4}}{x^{-3}} = x^{-4-(-3)} = x^{-1} = \dfrac{1}{x}$ $\dfrac{1}{x^{-3}} = \dfrac{x^0}{x^{-3}} = x^{0-(-3)} = x^3$ |
If the bases in a quotient match, subtract the exponents. If the bases do not match the rule does not apply. Notice that this rule follows from cancellation in the case of positive exponents. The rule does not apply to differences: $\qquad a^m-a^n \neq a^{m-n}$ |
|||||||||||
|
3. $\dfrac{1}{a^n} = a^{-n}$ (%%if $a \neq 0$) |
$\dfrac{1}{2^3} = 2^{-3}$
$5^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25}$ $\dfrac{1}{5^{-2}} = 5^{-(-2)} = 5^2 = 25$ |
#[See "Negative and zero exponents" above.][Ve "Exponentes negativos y cero " arriba.]#
#[Rule 3 is actually a special case of Rule 2:][Regla 3 es en realidad un caso especial de la Regla 2:]#
|
|||||||||||
| 4. $(a^n)^m = a^{nm}\ $ |
$(2^3)^2 = 2^{3\times 2} = 2^{6} = 64$
$(x^3)^4 = x^{3 \times 4} = x^{12}$ $(2^{-3})^2 = 2^{(-3)\times2} = 2^{-6} = \dfrac{1}{64}$ $(x^{-3})^{-4} = x^{(-3)\times(-4)} = x^{12}$ |
#[Raising a power to a power corresponds to mutliplying the powers.][Elevar una potencia a una potencia corresponde a multiplicar las potencias.]# |
Exponent laws to combine expressions with the different bases
| #[Rule][Regla]# | %%Examples | #[Comments][Comentarios]# | ||||||||
| 1. $(ab)^n = a^nb^n$ |
$(2 \cdot 3)^2 = 2^2 \cdot 3^2 = 4 \times 9 = 36$
$(2 \cdot 3)^{-2} = 2^{-2} \cdot 3^{-2} = \dfrac{1}{4} \times \dfrac{1}{9} = \dfrac{1}{36}$ $(4(-3))^{2} = 4^2 \cdot (-3)^2 = 16 \times 9 = 144$ $(xy)^{-4} = x^{-4}y^{-4}$ $(-xy)^3 = (-x)^3(y)^3$ |
The $n$th power of a product is the product of the $n$th powers. To see why this rule holds in the case of positive exponents, notice that
$\qquad (a+b)^n \neq a^n + b^n$ |
||||||||
|
2. $\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$ (%if $b \neq 0$) |
$\left(\dfrac{3}{2}\right)^4 = \dfrac{3^4}{2^4} = \dfrac{81}{16}$
$\left(\dfrac{x}{y}\right)^{-2} = \dfrac{x^{-2}}{y^{-2}}$ $\left(\dfrac{1}{y}\right)^3 = \dfrac{1^3}{y^3} = \dfrac{1}{y^3}$ $\left(\dfrac{-2}{-3}\right)^2 = \dfrac{(-2)^2}{(-3)^2} = \dfrac{4}{9}$ |
The $n$th power of a quotient is the quotient of the $n$th powers. To see why this rule holds in the case of positive exponents, notice that
$\qquad (a-b)^n \neq a^n - b^n$ |
#[Combining the identities][Combinando las identidades]#
Positive exponent form and power form
Consider, for example, the expression $\dfrac{2}{3x^2}.$ We rewrite this expression in various different ways using the exponent identities and the rule for multiplication of fractions:
$\dfrac{2}{3x^2}$ \t $=$ \t $\dfrac{2}{3} \cdot \dfrac{1}{x^2}$ \gap[40] \t #[Rule for multiplication of fractions][La regla para la multiplicación de fracciones]#
\\ \t $=$ \t $\dfrac{2}{3} \cdot x^{-2}$ \gap[40] \t #[Exponent identities Part 1 #3][Identidades del exponentes Parte 1 #3]#
$\dfrac{4x^{-5}}{7y}\ \ = \ \ \dfrac{4}{7} \cdot \dfrac{x^{-5}}{1} \cdot \dfrac{1}{y}\ \ = \ \ \dfrac{4}{7} \cdot \dfrac{1}{x^5} \cdot \dfrac{1}{y}\ \ = \ \ \dfrac{4}{7x^5y}$ \gap[40] \t #[Positive exponent form][Forma exponente positivo]#
\\ $\dfrac{4x^{-5}}{7y}\ \ = \ \ \dfrac{4}{7} \cdot \dfrac{x^{-5}}{1} \cdot \dfrac{1}{y}\ \ = \ \ \dfrac{4}{7} \cdot x^{-5} \cdot y^{-1}\ \ = \ \ \dfrac{4}{7}x^{-5}y^{-1}$ \gap[40] \t #[Power form][Forma potencia]#
#[Very important! Make sure you are comfortable with each step in these calculations!][¡Muy importante! ¡Asegúrate de que eres comodo con cada paso en estos cálculos!]#
#[Positive exponent form and power form ][Forma exponente positivo y forma potencia]#
#[An expression in positive exponent form is one in which all exponents are positive.][Una expresión de la forma exponente positivo es una en la que todos los exponentes son positivos.]#
#[Expressions in positive exponent form are often written using fractions with powers of variables in the numerator and/or the denominator.][Expresiones en la forma exponente positivo son frecuentemente escritas usando fracciones con potencias de variables en el numerador y/o el denominador.]#
%%Examples #[of expressions in positive exponent form][de expresiones de la forma exponente positivo]#
$\dfrac{3z^2}{4y^5} \qquad$ $\dfrac{2}{4x^3} \qquad$ $3.5z^8 \qquad$ $\dfrac{1}{x} \qquad$ $\dfrac{3}{4} \qquad$ $\dfrac{2}{x} - \dfrac{4x^3}{z}$
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 $ $x^{-1} \qquad$ $3x^0 \qquad$ $3^{-2} $
An expression in power form is one in which 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][Término con una variable; $a$ = constante, $x$ = variable, $n$ = cualquiera potencia]#
\\ $ax^my^n$ $\qquad$ \t #[Term with two variables; $a$ = constant, $x, y$ = variables, $n, m$ = any powers][Término con dos variables; $a$ = constante, $x, y$ = variables, $m, n$ = cualquieras potencias]#
\\ $ax^my^nz^k$ $\qquad$ \t #[Term with three variables; $a$ = constant, $x, y, z$ = variables, $n, m, k$ = any powers][Término con tres variables; $a$ = constante, $x, y, z$ = variables, $m, n, k$ = cualquieras potencias]#
%%Examples #[of expressions in power form][de expresiones de la forma potencia]#
$4z^{-2} \qquad$ $\dfrac{2}{3}x^{-1} \qquad$ $3 + x - x^2 \qquad$ $3x^2y^{-2} \qquad$ $4z^{-2} - 2y^{1/2}$
#[The following expressions are not in power form because they contain variables that appear in fractions:][Las siguientes expresiones no son de la forma potencia porque contienen variables que aparecen en fracciones:]#
$\dfrac{3x}{4} \qquad$ $\dfrac{3y^{-2}}{y} \qquad$ $\dfrac{2}{4y^{-3}} \qquad$ $y + \dfrac{1}{y} \qquad$ $\dfrac{2}{3x^{-1}}$
Converting to positive exponent or power form
We can use the exponent identities to convert expressions to one or other of the two forms just described:
Now try the exercises in Section 0.2 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