Section0.0Preliminaries
¶
Inequalities and interval notation.
Cartesian coordinate system.
Mathematical expressions with variables.
Subsection0.0.1Sets of Numbers
¶Before introducing functions, we'll need a brief summary of set theory and some of the associated vocabulary and notations we use in the text. We begin with a definition.
A set is a collection of objects which are called the elements of the set.
For example, the collection of letters that make up the word “coyote” is a set. Here are three ways to describe sets:
Ways to Describe a Set
The Verbal Method: Use a sentence to define the set.
The Roster Method: Begin with a left brace ' \(\{\)', list each element of the set once, end with a right brace ` \(\}\)'.
The Set-Builder Method: A combination of the verbal and roster methods using a mathemaical expression to describe the set.
Example0.0.1Describing Sets
-
For example, let \(S\) be the set described verbally as: The set of letters that make up the word “coyote”.
A roster description of \(S\) would be \(\{c,o,y,t,e\}\text{.}\) Note that we listed '\(o\)' only once, even though it appears twice in “coyote”. The order of the elements doesn't matter, so \(\{t,e,y,o,c\}\) also is a roster description of \(S\text{.}\)
-
Now consider the set, \(T\) of integers \(\{1,2,3,4,5,6,7,8,9\}\)
A set-builder description of \(T\) is: \(\{x \; | \; x\) is a positive integer with \(x /lt 10
\}\text{.}\) The way to read this is: “The set of positive integers \(x\) such that \(x\) is less than \(10\text{.}\)”
In each of the above cases, we may use the familiar equals sign '\(=\)' and write \(S = \{c,o,t,e,y\}\) or \(T = \{x \; | \; x \) is a positive integer with \(x \lt 10 \}\text{.}\) Clearly \(t\) is in \(S\) and \(3\) is not in \(S\text{.}\) Here are several famous sets of numbers.
Sets of Numbers
The Empty Set: \(\emptyset = \{ \}\text{.}\) This is the set with no elements. Like the number \(0,\) it plays a vital role in mathematics.
The Natural Numbers: \(\mathbb{N} = {1, 2, 3, \dots }\) The three periods in a row here indicate that the natural numbers contain \(1\text{,}\) \(2\text{,}\) \(3\text{,}\) and so forth.
The Integers: \(\mathbb{Z} = {\cdots ,-3, -2, -1, 0, 1, 2, 3, \cdots}\)
-
The Rational Numbers: \(\mathbb{Q} = \{\frac{a}{b}\; |\; a \mbox{ and } b \mbox{ are integers with } b \neq 0\}.\) Rational numbers are the ratios of integers (provided the denominator is not zero!). It turns out that another way to describe the rational numbers is:
\(\mathbb{Q} = \{x \; | \; x \mbox{ is a real number that has a repeating or terminating decimal representation}\}.\)
The Real Numbers: \(\mathbb{R} = \{ x \; | \; x \mbox{ is a decimal number}\}.\)
The Irrational Numbers: \(\mathbb{P} = \{ x \; | \; x \mbox{ is a non-rational real number }\}.\) Said another way, an irrational number is a decimal that is not a repeating decimal.
It is important to note that every natural number is an integer. Each integer is a rational number (taking \(b = 1\) in the above definition for \(\mathbb{Q}\) produces integers) and the rational numbers are all real numbers, since they possess decimal representations. For the most part, we will focus on sets whose elements come from the real numbers \(\mathbb{R}\text{.}\)
Subsection0.0.2Interval Notation
¶“Interval notation” is a shorthand way of describing an interval. For example, for real numbers \(a\) and \(b\text{,}\) “(a,b)” is interval notation for “the set of all real numbers \(x\) bigger than \(a\) and less than \(b\)”. The table given below displays interval notation for different kinds of intervals.
Region on the Real Number Line
All Real numbers, \(\mathbb{R}\)
Square brackets “\([ \)” or “\(] \)” indicate that an endpoint is included in the interval; round parentheses “\(( \)” or “\() \)” indicate that an endpoint is not included.
WeBWorK: Entering Infinity and Inequalities
Type
-
inf for \(\infty\) and -inf for \(-\infty\)
-
>= for \(\geq\) and <= for \(\leq\)
Note that we could write the inequality \(1 < x < 3\) as \(3 > x > 1\) and still have the same set. An easy way to remember which inequality symbol to use is that the smaller (pointy) end of the inequality sign points to the smaller number and conversely the wide end is next to the larger value.
Example0.0.2Describing Specific Sets
Region on the Real Number Line
We will need to combine sets. Intersection and union are two basic ways to combine sets. We define both of these concepts below.
Suppose \(A\) and \(B\) are two sets.
The intersection of \(A\) and \(B\) contains the elements that are in BOTH sets: \(A \cap B = \{ x \; | \; x \mbox{ is in } A \mbox{ and } x \mbox{ is in } B\}.\)
The union of \(A\) and \(B\) contains every element that occurs in AT LEAST ONE of the sets: \(A \cup B = \{ x \; | \; x \mbox{ is in } A \mbox{ or } x \mbox{ is in } B \mbox{ (or both)}\}.\)
Said differently, the intersection of two sets is the overlap of the two sets-all the elements which the sets both have in common. The union of two sets consists of the elements in both of the sets, collected together.
Example0.0.3Set Intersection and Union
For \(A = \{1, 2, 3\}\) and \(B = \{2, 4, 6\},\) we have \(A \cap B = \{ 2 \}\) and \(A \cup B = \{1, 2, 3, 4, 6\}.\)
Example0.0.4Interval Intersection and Union
If \(A = [-5, 3)\) and \(B=(1,\infty),\) then we can find \(A \cap B\) and \(A \cup B\) graphically.
First, sketch the graphs of \(A\) and \(B\text{.}\)
To determine \(A \cap B,\) we shade the overlap of the two and obtain \(A \cap B = (1, 3).\)
To determine \(A \cup B,\) we shade each of \(A\) and \(B\) along the lower number line and describe the resulting shaded region to see \(A \cup B = [-5, \infty).\)
Here are slightly more complicated situations where the set is in two separate pieces.
Example0.0.5More Intervals
Express the following sets of numbers using interval notation:
Region on the Real Number Line
\(x \leq -2\) or \(x \geq 2\)
\((-\infty, -2] \cup [2, \infty)\)
\((-\infty, 3)\cup (3, \infty)\)
WeBWorK: Entering Union \(\cup\) and Not Equal \(\neq\)
Type
- Upper case
U for union \(\cup\text{.}\)
-
!= for \(\neq\text{.}\)
Subsection0.0.3The Cartesian Coordinate Plane
¶Math became far more intuitive and powerful when people learned to marry the precision of algebra to the visual intuition of geometry. Simply put, we seek to find a way to draw algebraic things. Let's start with the Cartesian Coordinate Plane, named in honor of the 17th century philosopher, mathematician and scientist, Rene Decartes.
Imagine two real number lines crossing at a right angle at \(0\) as drawn below.
The horizontal number line is usually called the \(x\)-axis while the vertical number line is usually called the \(y\)-axis. As with the usual number line, we imagine these axes extending off indefinitely in both directions. Having two number lines enables us to locate the positions of points off of the number lines as well as points on the lines themselves.
For example, consider the point \(P\) shown below:
To use the numbers on the axes to label this point, imagine dropping a vertical line from the \(x\)-axis to \(P\) and extending a horizontal line from the \(y\)-axis to \(P\text{.}\) This process is sometimes called “projecting” the point \(P\) to the \(x\)- (respectively \(y\)-) axis. We then locate the point \(P\) using the ordered pair \((2,-4).\) The first number in the ordered pair is the \(x\)-coordinate and the second is the \(y\)-coordinate. Taken together, the ordered pair \((2, -4)\) gives the position of the point \(P\) with respect to the \(x\) and \(y\) axes.
Notice that the order in the ordered pair is important: if we wish to plot the point \((-4, 2),\) we would move to the left \(4\) units from the origin and then move upwards \(2\) units.
The axes divide the plane into four regions called quadrants. They are labeled \(Q1, Q2, Q3\) and \(Q4\) and proceed counterclockwise around the plane as shown in the graph below. Note that point \(P\) is in quadrant 4, \(Q4,\) and \(Q\) is in quadrant 2, \(Q2.\)
Subsection0.0.4Mathematical expressions
¶You should have be able to simplify basic mathematical expressions with variables and know how to solve an equation for a variable. Below are a few examples for your review.
Example0.0.6Evaluate an Expression
Evaluate \(2p + \dfrac{r^2}{3}\) for \(p = -1\) and \(r = 4\text{.}\)
Solution: Substitute \(-1\) for \(p\) and \(4\) for \(r\text{,}\) and simplify:
\begin{equation*}
2(-1) + \frac{4^2}{3} = -2 + \frac{16}{3} = \frac{-6}{3} + \frac{16}{3} = \frac{10}{3}\checkmark
\end{equation*}
Example0.0.7Simplify an Expression
Combine like terms: \(3(2x+5)-4x\text{.}\)
In this case one may remove the parentheses by multiplying out the expression \(3(2x+5)\) so that the like terms may be combined.
\begin{gather*}
3(2x+5)-4x\\
3(2x)+3(5)-4x\\
6x+15-4x\\
(6x-4x)+15\\
2x+15 \checkmark
\end{gather*}
Example0.0.8Solve an Equation
Solve for \(x\text{:}\) \(\dfrac{x}{3}+2=\dfrac{x}{4}\)
There are many ways to start this problem. If you don't like fractions you may "clear" the denominators by multiplying through both sides of the equation by \(12\text{.}\) Then proceed to combine the \(x\)'s on one side.
\begin{align*}
12\left(\frac{x}{3} + 2\right) & = 12\left(\frac{x}{4}\right)\\
12\left(\frac{x}{3}\right) + 12(2) & = 3x\\
4x + 24 &= 3x\\
4x + 24 - 3x & = 3x-3x\\
x + 24 & = 0\\
x & = -24\checkmark
\end{align*}
Example0.0.9Solve an Equation
Solve for \(C\text{:}\) \(S = C + rC\)
The equation needs to be rearranged so that \(C\) is isolated. This can be done by factoring \(C\) out of the right side then dividing by \(1+r\text{:}\)
\begin{gather*}
S = C + rC\\
S = C(1 + r)\\
\frac{S}{1+r} = C
\end{gather*}
Which may be rewritten in the form: \(C = \dfrac{S}{1+r} \checkmark\)
Subsection0.0.5Exponent Properties
¶Problems with exponents may often be simplified using a few basic exponent properties. For now, exponents represent repeated multiplication. We will use this fact to discover the important properties.
World View Note
The word exponent comes from the Latin “expo” meaning out of and “ponere” meaning place. While there is some debate, it seems that the Babylonians living in Iraq were the first to do work with exponents (dating back to the 23rd century BC or earlier!).
Example0.0.10Multiply \(a^n\) times \(a^m\) using Exponents
Multipy \(a^3\) times \(a^2\text{:}\)
\begin{align*}
a^3a^2 \amp\amp\amp \text{Represent }a^4\text{ and }a^2\text{ as products}\\
(aaa)(aa) \amp\amp\amp \text{Regroup}\\
(aaaaa) \amp\amp\amp \text{Represent the product with an exponent}\\
a^5\amp\amp\amp\text{Our Solution} \checkmark
\end{align*}
The example shows that \(a^3\) times \(a^2\) just represents \(3+2\) \(a\)'s multiplied together. Similarly, \((a^m)(a^n)\) just represents \((m+n)\) \(a\)'s multiplied together. This is known as the product rule of exponents:
\begin{gather}
\text{Product Rule of Exponents: }
a^m a^n=a^{m+n}\label{exponents-product-rule}\tag{0.0.1}
\end{gather}
Note: In the expression \(a^n\text{,}\) \(n\) is called the exponent and \(a\) is called the base.
WeBWorK: Entering Exponents
Type a^5 for \(a^5.\)
The product rule of exponents can be used to simplify many problems. This is shown in the following examples.
Example0.0.11Product Rule of Exponents
Multiply:
\begin{align*}
3^2\cdot 3^6\cdot 3 \amp\amp\amp \text{Same base \((3)\), add the exponents \(2+6+1\)}\\
3^9 \amp\amp\amp \text{Our Solution}\checkmark
\end{align*}
Example0.0.12Product Rule of Exponents
Multiply:
\begin{align*}
2x^3 y^5 z \cdot 5x y^2 z^3 \amp\amp\amp \text{Multiply \(2\cdot 5\), add exponents on each of \(x, y\) and \(z\)}\\
10x^4 y^7 z^4 \amp\amp\amp \text{Our Solution}\checkmark
\end{align*}
Next, we consider what happens when we divide with exponents.
Example0.0.13Division with Exponents
Divide:
\begin{align*}
\dfrac{a^5}{a^2} \amp\amp\amp \text{Expand exponents}\\
\dfrac{aaaaa}{aa} \amp\amp\amp \text{Since \(\frac{a}{a} = 1\), each \(a\) in the bottom "cancels" an \(a\) in the top.}\\
aaa \amp\amp\amp \text{Convert to exponents}\\
a^3 \amp\amp\amp \text{Our Solution}\checkmark
\end{align*}
Because the two \(a\)''s represented by \(a^2\) cancel two of the five \(a\)'s represented by \(a^5\text{,}\) there are \(5-2=3\) \(a\)'s left after cancelling. Thus, dividing \(a^5\) by \(a^2\) just amounts to subtracting the exponents \(\dfrac{a^5}{a^2} = a^{5-2}= a^3\text{.}\) In a similar way, \(\dfrac{a^m}{a^n} = a^{m-n}\text{.}\) This is known as the quotient rule of exponents:
\begin{gather}
\text{Quotient Rule of Exponents: }
\frac{a^m}{ a^n} =a^{m-n}\label{exponents-quotient-rule}\tag{0.0.2}
\end{gather}
Example0.0.14Quotient Rule of Exponents
Divide:
\begin{align*}
\dfrac{7^5 x^8}{7^4 x^3} \amp\amp\amp \text{Subtract the exponents on like bases, \(5-4 = 1\) and \(8-3=5\)}\\
7^1x^5 \amp\amp\amp \text{Note that \(7^1 = 7\) since there is only one \(7\) to multiply.}\\
7x^5 \amp\amp\amp \text{Our Solution}\checkmark
\end{align*}
Example0.0.15Quotient Rule of Exponents
Simplify:
\begin{align*}
\dfrac{5a^3 b^5 c^2}{2ab^3 c} \amp\amp\amp \text{Subtract exponents on \(a, b\) and \(c\)}\\
\frac{5}{2} a^2 b^2 c\amp\amp\amp \text{Our Solution}\checkmark
\end{align*}
A third property we will look at will have an exponent problem raised to a second exponent. This is investigated in the following example:
Example0.0.16Powers with Exponents
Multiply:
\begin{align*}
\left(a^2\right)^3 \amp\amp\amp \text{This means we have three \(a^2\)'s multiplied together}\\
a^2 \cdot a^2 \cdot a^2 \amp\amp\amp \text{Each }a^2\text{ represent the produce of two }a\text{'s}\\
(aa)(aa)(aa) \amp\amp\amp \text{Three groups of two make six }a\text{'s in all multiplied together}\\
a^6 \amp\amp\amp \text{Our Solution}\checkmark
\end{align*}
Since three groups of two is \(3 \cdot 2 = 6\text{,}\) a quicker method to arrive at the solution is to just multiply the exponents, \((a^2)^3 = a^{2\cdot 3} = a^6\text{.}\) This is known as the power of a power rule of exponents:
\begin{gather}
\text{Power of a Power Rule of Exponents: }
(a^m)^n=a^{mn}\label{exponents-power-of-a-power-rule}\tag{0.0.3}
\end{gather}
This property is often combined with two other properties which we will investigate now.
Example0.0.17Powers of Products
Multiply:
\begin{align*}
(ab)^3 \amp\amp\amp \text{This means we have \((ab)\) three times}\\
(ab)(ab)(ab) \amp\amp\amp \text{Regroup the variables}\\
(aaa)(bbb) \amp\amp\amp \text{The product of three \(a\)'s and three \(b\)'s can be written with exponents}\\
a^3 b^3 \amp\amp\amp \text{Our Solution}\checkmark
\end{align*}
A quicker method to arrive at the solution is to take the exponent, \(3,\) and put it on each factor in parentheses, \((ab)^3 = a^3b^3\text{.}\) This is known as the power of a product rule of exponents:
\begin{gather}
\text{Power of a Product Rule of Exponents: }
(ab)^m=a^m b^m\label{exponents-power-of-a-product-rule}\tag{0.0.4}
\end{gather}
It is important to be careful to only use the power of a product rule with multiplication inside parentheses. This property does NOT work if there is addition or subtraction inside the parentheses.
WARNING:
The Power of a Product Rule only applies to products. It does not apply to sums for any exponent not equal to \(1\text{.}\) For example,
\begin{align*}
(1+2)^2 = 3^2 = 9 \amp\amp \text{But,}\\
1^2+2^2 = 1+4 = 5 \amp\amp \text{So, }\\
(1+2)^2 \neq 1^2+2^2 \amp\amp \text{The rule does not hold for a sum.}
\end{align*}
\begin{align*}
(a+b)^m\ne a^m+b^m \amp\amp\amp \text{These are NOT equal if \(m\ne 1\), beware of this error!}
\end{align*}
Another property that is very similar to the power of a product rule is considered next.
Example0.0.18Powers of Quotients
Divide:
\begin{align*}
\left(\frac{a}{b} \right)^3 \amp\amp\amp \text{This means we have the three copies of the fraction multiplied together}\\
\left(\frac{a}{b} \right) \left(\frac{a}{b} \right) \left(\frac{a}{b} \right) \amp\amp\amp \text{Regroup the bases}\\
\frac{aaa}{bbb} \amp\amp\amp \text{Multiply fractions across the top and bottom}\\
\dfrac{a^3}{b^3} \amp\amp\amp \text{Our Solution}\checkmark
\end{align*}
A quicker method to arrive at the solution is to put the exponent on every factor in both the numerator and denominator, \(\left(\frac{a}{b} \right)^3=\frac{a^3}{b^3}\text{.}\) This is known as the power of a quotient rule of exponents:
\begin{gather}
\text{Power of a Quotient Rule of Exponents: }
\left(\frac{a}{b} \right)^m=\frac{a^m}{b^m}\label{exponents-power-of-a-quotient-rule}\tag{0.0.5}
\end{gather}
The power of a power, product and quotient rules are often used together to simplify expressions. This is shown in the following examples:
Example0.0.19Simplify Exponents
Simplify.
\begin{align*}
(x^3y)^4 \amp\amp\amp \text{Apply the exponent }4\text{ to each factor}\\
(x^3)^4 y^4 \amp\amp\amp \text{Multiply powers}\\
x^{12} y^4 \amp\amp\amp \text{Our Solution}\checkmark
\end{align*}
WeBWorK: Entering Multi-digit Exponents
If your exponent has two or more digits, you must place parentheses around it. Otherwise, WeBWorK only takes the first number after the ^ as the exponent: Type x^(12) for \(x^{12}\text{.}\)
Example0.0.20Simplify Exponents
Simplify.
\begin{align*}
\left(\dfrac{a^3 b}{c^4 d^5} \right)^2 \amp\amp\amp \text{ }\\
\dfrac{(a^3)^2 b^2}{(c^4)^2 (d^5)^2} \amp\amp\amp \text{ }\\
\dfrac{a^6 b^2}{c^8d^{10}} \amp\amp\amp \text{Our Solution}\checkmark
\end{align*}
An expression such as \(5^3\) does not mean we multiply \(5\) by \(3\text{,}\) rather we multiply three \(5\)'s together, \(5 \cdot 5 \cdot 5 = 125\text{.}\) This is shown in the next example.
Example0.0.21Simplify Exponents
Simplify.
\begin{align*}
(4x^2 y^5)^3 \amp\amp\amp \text{}\\
4^3 (x^2)^3 (y^5)^3 \amp\amp\amp \text{}\\
4^3 x^6 y^{15} \amp\amp\amp \text{Evaluate \(4^3=4\cdot 4\cdot 4\)}\\
64x^6 y^{15} \amp\amp\amp \text{Our Solution}\checkmark
\end{align*}
Never multiply a base by the exponent. In the previous example we did not put the \(3\) on the \(4\) and multipy to get \(12\text{,}\) this would have been incorrect. These properties pertain to exponents only, not bases.
In this lesson we have discussed five different exponent properties. These rules are summarized in the following table:
| Rules of Exponents |
| Product Rule of Exponents |
\(a^m a^n=a^{m+n}\)(0.0.1)
|
| Quotient Rule of Exponents |
\(\frac{a^m}{a^n} =a^{m-n}\)(0.0.2)
|
| Power of a Power Rule of Exponents |
\((a^m)^n=a^{mn}\)(0.0.3)
|
| Power of a Product Rule of Exponents |
\((ab)^m=a^m b^m\)(0.0.4)
|
| Power of a Quotient Rule of Exponents |
\(\left(\frac{a}{b} \right)^m=\frac{a^m}{b^m}\)(0.0.5)
|
These five properties are often mixed up in the same problem. Often there is a bit of flexibility as to which property is used first. However, order of operations still applies to a problem. For this reason it is our suggestion to simplify inside any parentheses first, then simplify any exponents (using power rules), and finally simplify any multiplication or division (using product and quotient rules). This is illustrated in the next few examples.
Example0.0.22Simplify Exponents
Simplify.
\begin{align*}
(4x^3 y \cdot 5x^4 y^2)^3 \amp\amp\amp \text{In parenthesis simplify using Product Rule, add exponents}\\
(20x^7 y^3)^3 \amp\amp\amp \text{Apply the Power of a Product Rule, multiply exponents}\\
20^3x^{21}y^9)^3\amp\amp\amp \text{Evaluate }20^3\\
8000x^{21} y^9 \amp\amp\amp \text{Our Solution}\checkmark
\end{align*}
Example0.0.23Simplify Exponents
Simplify.
\begin{align*}
7a^3(2a^4)^3 \amp\amp\amp \text{Parentheses are already simplified, next use Power Rule}\\
7a^3(8a^{12})\amp\amp\amp \text{Using Product Rule, add exponents and multiply numbers}\\
56a^{15} \amp\amp\amp \text{Our Solution}\checkmark
\end{align*}
Example0.0.24Simplify Exponents
Simplify.
\begin{align*}
\dfrac{(3a^3 b)(10a^4 b^3)}{2a^4 b^2} \amp\amp\amp \text{Simplify numerator with Product Rule, adding exponents}\\
\dfrac{30a^7 b^4}{2a^4 b^2}\amp\amp\amp \text{Now use the Quotient Rule, subtract exponents}\\
15a^3b^2 \amp\amp\amp \text{Our Solution}\checkmark
\end{align*}
These problems can quickly become quite involved. Remember to follow order of operations as a guide, simplify inside parenthesis first, then power rules, then product and quotient rules.
Subsection0.0.6Negative Exponents
A few exponent properties deal with exponents that are not positive. The first is considered in the following example, which is worked out two different ways:
Example0.0.25Investigate Zero Power
\begin{align*}
\text{ } \amp\amp\amp \underline{\text{First Method}}\amp\amp\amp \text{ } \amp\amp\amp \underline{\text{Second Method}}\\
\dfrac{a^3}{a^3} \amp\amp\amp \text{Use the Quotient Rule,}\amp\amp\amp \dfrac{a^3}{a^3} \amp\amp\amp \text{Rewrite exponents}\\
\text{ } \amp\amp\amp \text{subtract exponents}\amp\amp\amp \text{ } \amp\amp\amp \text{as repeated multiplication}\\
a^0 \amp\amp\amp \text{Method 1 Solution}\amp\amp\amp \dfrac{aaa}{aaa} \amp\amp\amp \text{Cancel out all the \(a\)'s}\\
\text{ } \amp\amp\amp \text{ }\amp\amp\amp \frac{1}{1}=1 \amp\amp\amp \text{Method 2 Solution}
\end{align*}
When we combine the two solutions we get:
\begin{align*}
\text{ } \amp\amp\amp a^0=1 \amp\amp\amp \text{ }
\end{align*}
This result is an important property known as the Zero Power Rule of Exponents:
\begin{gather}
\text{Zero Power Rule of Exponents: }
a^0=1\label{exponents-zero-power-rule}\tag{0.0.6}
\end{gather}
Any number (except \(0\)) or expression raised to the zero power will always be \(1\text{.}\) This is illustrated in the following example.
Example0.0.26Zero Power Rule of Exponents
Simplify.
\begin{align*}
(3x^2)^0 \amp\amp\amp \text{Apply the Zero Power Rule}\\
1 \amp\amp\amp \text{Our Solution}\checkmark
\end{align*}
Another property we will consider here deals with negative exponents. Again we will solve the following example two ways.
Example0.0.27Negative Exponents
\begin{align*}
\text{ } \amp\amp\amp \underline{\text{First Method}}\amp\amp\amp \text{ } \amp\amp\amp \underline{\text{Second Method}}\\
\dfrac{a^3}{a^5} \amp\amp\amp \text{Using the Quotient Rule,}\amp\amp\amp \dfrac{a^3}{a^5} \amp\amp\amp \text{Rewrite exponents}\\
\text{ } \amp\amp\amp \text{subtract exponents \(3-5=-2\)}\amp\amp\amp \text{ } \amp\amp\amp \text{as repeated multiplication}\\
a^{-2} \amp\amp\amp \text{Method 1 Solution}\amp\amp\amp \dfrac{aaa}{aaaaa} \amp\amp\amp \text{Reduce three \(a\)'s out of top and bottom}\\
\text{ } \amp\amp\amp \text{ } \amp\amp\amp \dfrac{1}{aa} \amp\amp\amp \text{Simplify to exponents}\\
\text{ } \amp\amp\amp \text{ } \amp\amp\amp \dfrac{1}{a^2} \amp\amp\amp \text{Method 2 Solution}
\end{align*}
Both expressdions represent the same thing so they must be equal:
\begin{align*}
\text{} \amp\amp\amp a^{-2}=\dfrac{1}{a^2} \amp\amp\amp \text{}
\end{align*}
This example illustrates an important property of exponents.
\begin{gather}
\text{Negative Power Rule of Exponents: }
a^{-n}= \frac{1}{a^n} \text{ if } a \neq 0\label{exponents-negative-power-rule}\tag{0.0.7}
\end{gather}
It is important to note that a negative exponent does not mean the expression is negative.
Example0.0.28Simplify a Fraction with a Negative Exponent
Simplify.
\begin{align*}
\left(\dfrac{5}{3} \right)^{-2} \amp\amp\amp \text{Negative exponents tell us to take the reciprocal}\\
\left(\dfrac{3}{5} \right)^2 \amp\amp\amp \text{The reciprocal has a positive exponent, now power rule}\\
\dfrac{3^2}{5^2} \amp\amp\amp \text{Evaluate}\\
\dfrac{9}{25} \amp\amp\amp \text{Our Solution}\checkmark
\end{align*}
In the example above, we simplified the reciprocal of the fraction by using the method for dividing fractions,
\begin{align*}
\frac{1}{\frac{a}{b}} = 1 \cdot \frac{b}{a} = \frac{b}{a} \amp\amp\amp \text{}
\end{align*}
Following is the rule for negative exponents:
| Rule of Negative Exponents |
| \(a^{-m}=\dfrac{1}{a^m}\) |
We now have the following nine properties of exponents. It is important that we are very familiar with all of them.
| Properties of Exponents |
| \(a^m a^n=a^{m+n}\) |
\((ab)^m=a^m b^m\) |
\(a^{-m}=\dfrac{1}{a^m}\) |
| \(\frac{a^m}{a^n}=a^{m-n}\) |
\(\left(\frac{a}{b} \right)^m=\frac{a^m}{b^m}\) |
\(\dfrac{1}{a^{-m}}= a^m\) |
| \((a^m)^n=a^{mn}\) |
\(a^0=1\) |
\(\left( \dfrac{a}{b}\right)^{-m}=\dfrac{b^m}{a^m}\) |
World View Note
Nicolas Chuquet, the French mathematician of the 15th century wrote \(12^{1\overline{m}}\) to indicate \(12x^{-1}\text{.}\) This was the first known use of the negative exponent.
