Using the Quotient Rule

Write each of the following products with a single base. Do not simplify further.

$\text{\textcircled{a}} \enspace \dfrac{(-2)^{14}}{(-2)^9}$ $\quad$ $\text{\textcircled{b}} \enspace \dfrac{t^{23}}{t^{15}}$ $\quad$ $\text{\textcircled{c}} \enspace \dfrac{\Big(z\sqrt{2}\Big)^5}{z\sqrt{2}}$

Solution

Use the quotient rule to simplify each expression.

$\text{\textcircled{a}} \enspace \dfrac{(-2)^{14}}{(-2)^9} = (-2)^{14-9} = (-2)^5$

$\text{\textcircled{b}} \enspace \dfrac{t^{23}}{t^{15}} = t^{23-15} = t^8$

$\text{\textcircled{c}} \enspace \dfrac{\Big(z\sqrt{2}\Big)^5}{z\sqrt{2}} = \Big(z\sqrt{2}\Big)^{5-1} = \Big(z\sqrt{2}\Big)^4$

Case

The above case, including the title and solution, is attributed to College Algebra 2e. Page 25-26.

Q&A

What are natural numbers?

Page 8, College Algebra 2e

The numbers we use for counting, or enumerating items, are the natural numbers: $1, 2, 3, 4, 5,$ and so on. We describe them in set notation as $\{1,2,3,\mathellipsis\}$ where the ellipsis $(\mathellipsis)$ indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers. Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers.

What are the set of whole numbers?

Page 8, College Algebra 2e

The set of whole numbers is the set of natural numbers plus zero: $\{0,1,2,3,\mathellipsis\}$ .

What are the set of integers?

Page 8, College Algebra 2e

The set of integers adds the opposites of the natural numbers to the set of whole numbers: $\{\mathellipsis,-3,-2,-1,0,1,2,3,\mathellipsis\}$ . It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.

$\stackrel{\text{negative integers}}{\mathellipsis,-3,-2,-1} \quad \stackrel{\text{zero}}{0} \quad \stackrel{\text{positive integers}}{1,2,3,\mathellipsis}$

What are the set of rational numbers?

Page 8, College Algebra 2e

The set of rational numbers is written as $\{\frac{m}{n} \text{ | } m \text{ and } n \text{ are integers and } n \not = 0 \}$ . Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.

Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:

• a terminating decimal:$\frac{15}{8} = 1.875$, or
• a repeating decimal: $\frac{4}{11} = 0.36363636\mathellipsis=0.\overline{36}$

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

What are irrational numbers?

Page 9, College Algebra 2e

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even $\frac{3}{2}$ but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers. Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown. $\{h \text{ | } h \text{ is not a rational number}\}$

What are real numbers?

Page 10, College Algebra 2e

Given any number n, we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers. As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.

The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one- to-one correspondence. We refer to this as the real number line as shown in Figure 1.

What are order of operations?

Page 12-13, College Algebra 2e

Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS:

• P(arentheses)
• E(xponents)
• M(ultiplication) and D(ivision)
• A(ddition) and S(ubtraction)

What are commutative properties?

Page 14, College Algebra 2e

The commutative property of addition states that numbers may be added in any order without affecting the sum. $a+b=b+a$

We can better see this relationship when using real numbers. $(-2)+7=5 \quad \text{and} \quad 7+(-2)=5$

Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product. $a \cdot b=b \cdot a$

Again, consider an example with real numbers. $(-11)\cdot(-4)=44 \quad \text{and} \quad (-4)\cdot(-11)=44$

It is important to note that neither subtraction nor division is commutative. For example, $17-5$ is not the same as $5-17$ Similarly, $20\div5 \not = 5\div20$.

What are associative properties?

Page 14-15, College Algebra 2e

The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same. $a(bc)=(ab)c$

Consider this example. $(3\cdot4)\cdot5=60 \quad \text{and} \quad 3\cdot(4\cdot5)=60$

The associative property of addition tells us that numbers may be grouped differently without affecting the sum. $a+(b+c)=(a+b)+c$

This property can be especially helpful when dealing with negative integers. Consider this example. $[15+(-9)]+23=29 \quad \text{and} \quad 15+[(-9)+23]=29$

Are subtraction and division associative? Review these examples.

$$\begin{align*} {4} 8-(3-15) &\stackrel{?}{=} (8-3)-15 &\quad 64\div(8\div4) &\stackrel{?}{=} (64\div8)\div4 \\ 8-(-12) &= 5-15 &\quad 64\div2 &\stackrel{?}{=} 8\div4 \\ 20 &\not= -10 &\quad 32 &\not= 2 \end{align*}$$

As we can see, neither subtraction nor division is associative.

What are distributive properties?

Page 15, College Algebra 2e

The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum. $a \cdot (b+c) = a \cdot b + a \cdot c$

This property combines both addition and multiplication (and is the only property to do so). Let us consider an example.

Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by –7, and adding the products.

To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.

$$\begin{align*} 6+(3\cdot5) &\stackrel{?}{=} (6+3)\cdot(6+5) \\ 6+(15) &\stackrel{?}{=} (9)\cdot(11) \\ 21 &\not= 99 \end{align*}$$

A special case of the distributive property occurs when a sum of terms is subtracted. $a-b=a+(-b)$

For example, consider the difference $12-(5+3)$. We can rewrite the difference of the two terms $12$ and $(5+3)$ by turning the subtraction expression into addition of the opposite. So instead of subtracting $(5+3)$ we add the opposite. $12+(-1)\cdot(5+3)$

Now, distribute $-1$ and simplify the result.

$$\begin{align*} 12-(5+3) &= 12+(-1)\cdot(5+3) \\ &= 12+[(-1)\cdot5+(-1)\cdot3] \\ &= 12+(-8) \\ &= 4 \end{align*}$$

This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.

$$\begin{align*} 12-(5+3) &= 12+(-5-3) \\ &= 12+(-8) \\ &= 4 \end{align*}$$

What are identity properties?

Page 15, College Algebra 2e

The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number. $a+0=a$

The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number. $a\cdot1=a$

For example, we have $(-6)+0=-6$ and $23\cdot1=23$. There are no exceptions for these properties; they work for every real number, including $0$ and $1$.

What are inverse properties?

Page 16, College Algebra 2e

The inverse property of addition states that, for every real number a, there is a unique number, called the additive inverse (or opposite), denoted by $(−a)$, that, when added to the original number, results in the additive identity, $0$. $a+(-a)=0$

For example, if $a=-8$, the additive inverse is $8$, since $(-8)+8=0$.

The inverse property of multiplication holds for all real numbers except $0$ because the reciprocal of $0$ is not defined. The property states that, for every real number a, there is a unique number, called the multiplicative inverse (or reciprocal), denoted $\frac{1}{a}$, that, when multiplied by the original number, results in the multiplicative identity, $1$. $a\cdot\frac{1}{a}=1$

For example, if $a=-\frac{2}{3}$, the reciprocal, denoted $\frac{1}{a}$, is $\frac{3}{2}$ because $a\cdot\frac{1}{a}=\Big(-\frac{2}{3}\Big)\cdot\Big(-\frac{3}{2}\Big)=1$

What are properties of real numbers?

Page 16, College Algebra 2e

The following properties hold for real numbers a, b, and c.

	Addition	Multiplication
Commutative Property	$a+b=b+a$	$a \cdot b=b \cdot a$
Associative Property	$a+(b+c)=(a+b)+c$	$a(bc)=(ab)c$
Distributive Property	$a\cdot(b+c)=a \cdot b+a \cdot c$	$a\cdot(b+c)=a \cdot b+a \cdot c$
Identity Property	There exists a unique real number called the additive identity, $0$, such that, for any real number $a$ $a+0=a$	There exists a unique real number called the multiplicative identity, $1$, such that, for any real number $a$ $a\cdot1=a$
Inverse Property	Every real number $a$ has an additive inverse, or opposite, denoted $-a$, such that $a+(-a)=0$	Every nonzero real number $a$ has a multiplicative inverse, or reciprocal, denoted $\frac{1}{a}$, such that $a \cdot \Big(\frac{1}{a}\Big)=1$

Evaluating Algebraic Expressions

Page 17, College Algebra 2e

So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as $x+5,\frac{4}{3} \pi r^3$, or $\sqrt{2m^3n^2}$ . In the expression $x+5$, $5$ is called a constant because it does not vary and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.

We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.

$$\begin{align*} {3} (-3)^5 &= (-3)\cdot(-3)\cdot(-3)\cdot(-3)\cdot(-3) &\quad x^5 &= x \cdot x \cdot x \cdot x \cdot x \\ (2\cdot7)^3 &= (2\cdot7) \cdot (2\cdot7) \cdot (2\cdot7) &\quad (yz)^3 &= (yz)\cdot(yz)\cdot(yz) \\ \end{align*}$$

In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.

Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.

What is an equation?

Page 19, College Algebra 2e

An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation $2x+1=7$ has the solution $3$ because when we substitute $3$ for $x$ in the equation, we obtain the true statement $2(3)+1=7$.

What is a formula?

Page 19, College Algebra 2e

A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area $A$ of a circle in terms of the radius $r$ of the circle: $A=\pi r^2$. For any value of $r$, the area $A$ can be found by evaluating the expression $\pi r^2$.

How is the product rule of exponents used?

Page 24, College Algebra 2e

Consider the product $x^3\cdot x^4$. Both terms have the same base, $x$, but they are raised to different exponents. Expand each expression, and then rewrite the resulting expression.

$$\begin{align*} x^3\cdot x^4 \enspace &= \enspace \stackrel{3 \text{ factors}}{x\cdot x\cdot x} \cdot \stackrel{4 \text{ factors}}{x\cdot x\cdot x\cdot x} \\ &= \enspace \stackrel{7 \text{ factors}}{x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x} \\ &= \enspace x^7 \end{align*}$$

The result is that $x^3\cdot x^{3+4} = x^7$.

Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the product rule of exponents.

$$a^m\cdot a^n = a^{m+n}$$

Now consider an example with real numbers.

$$2^3\cdot 2^4 = 2^{3+4} = 2^7$$

We can always check that this is true by simplifying each exponential expression. We find that $2^3$ is $8$, $2^4$ is $16$, and $2^7$ is $128$. The product $8\cdot 16$ equals $128$, so the relationship is true. We can use the product rule of exponents to simplify expressions that are a product of two numbers or expressions with the same base but different exponents.

How is the quotient rule of exponents used?

Page 25, College Algebra 2e

The quotient rule of exponents allows us to simplify an expression that divides two numbers with the same base but different exponents. In a similar way to the product rule, we can simplify an expression such as $\dfrac{y^m}{y^n}$, where $m>n$. Consider the example $\dfrac{y^9}{y^5}$. Perform the division by canceling common factors.

$$\begin{align*} \dfrac{y^9}{y^5} &= \dfrac{y\cdot y\cdot y\cdot y\cdot y\cdot y\cdot y\cdot y\cdot y}{y\cdot y\cdot y\cdot y\cdot y} \\ &= \dfrac{\cancel{y}\cdot \cancel{y}\cdot \cancel{y}\cdot \cancel{y}\cdot \cancel{y}\cdot y\cdot y\cdot y\cdot y}{\cancel{y}\cdot \cancel{y}\cdot \cancel{y}\cdot \cancel{y}\cdot \cancel{y}} \\ &= \dfrac{y\cdot y\cdot y\cdot y}{1} \\ &= y^4 \end{align*}$$

Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend.

$$\dfrac{a^m}{a^n} = a^{m-n}$$

In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents.

$$\dfrac{y^9}{y^5} = y^{9-5} = y^4$$

For the time being, we must be aware of the condition $m>n$. Otherwise, the difference $m-n$ could be zero or negative. Those possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify matters and assume from here on that all variables represent nonzero real numbers.

Glossary

Associative Property

https://en.wikipedia.org/wiki/Associative_property

The associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result.

Consider the following equations:

$$\begin{aligned} (2+3)+4 &= 2+(3+4) = 9 \\ 2\times (3\times 4) &= (2\times 3)\times 4 = 24 \end{aligned}$$

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Commutative Property

https://en.wikipedia.org/wiki/Commutative_property

A binary operation is commutative if changing the order of the operands does not change the result.

E.g. $3 + 4 = 4 + 3$ or $2 × 5 = 5 × 2$

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Decimal

https://en.wikipedia.org/wiki/Decimal

A decimal numeral (also often just decimal or, less correctly, decimal number), refers generally to the notation of a number in the decimal numeral system. Decimals may sometimes be identified by a decimal separator (usually "." or "," as in 25.9703 or 3,1415).

The decimal numeral system (also called the base-ten positional numeral system and denary /ˈdiːnəri/[1] or decanary) is the standard system for denoting integer and non-integer numbers.

Denominator

https://en.wiktionary.org/wiki/denominator

The number or expression written below the line in a fraction (such as 2 in ½).

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Distributive Property

https://en.wikipedia.org/wiki/Distributive_property

The distributive property of binary operations is a generalization of the distributive law, which asserts that the equality

$$x\cdot (y+z) = x\cdot y + x\cdot z$$

is always true in elementary algebra. For example, in elementary arithmetic, one has

$$2\cdot (1+3) = 2\cdot 1 + 2\cdot 3$$

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Equation

https://en.wikipedia.org/wiki/Equation

An equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign $=$.

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Expression

https://en.wikipedia.org/wiki/Expression_(mathematics)

An expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.

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Fraction

https://en.wikipedia.org/wiki/Fraction

A fraction represents a part of a whole or, more generally, any number of equal parts.

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Identity Property

https://simple.wikipedia.org/wiki/Identity_Property

In math, the identity property is made up of two parts: the additive identity property and the multiplicative identity property.

The additive identity property says that the sum of adding any number and zero (0) is just the original number. For that reason, zero is often called the additive identity of common numbers.

The multiplicative identity property says that the product of multiplying any number and one (1) is just the original number. Also, if you divide a number by itself, the result (quotient) is one. For that reason, one is often called the multiplicative identity of common numbers.

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Integer

https://en.wikipedia.org/wiki/Integer

An integer is the number zero (0), a positive natural number (1, 2, 3, etc.) or a negative integer (−1, −2, −3, etc.).

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Inverse Property

https://en.wikipedia.org/wiki/Additive_inverse

https://en.wikipedia.org/wiki/Multiplicative_inverse

The additive inverse of a number $a$(sometimes called the opposite of $a$) is the number that, when added to $a$, yields zero.

A multiplicative inverse or reciprocal for a number $x$, denoted by $\frac{1}{x}$ or $x^{−1}$, is a number which when multiplied by $x$ yields the multiplicative identity, $1$

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Irrational Number

https://en.wikipedia.org/wiki/Irrational_number

All the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers.

Among irrational numbers are the ratio $\pi$ of a circle's circumference to its diameter, Euler's number $e$, the golden ratio $\phi$, and the square root of two.[1] In fact, all square roots of natural numbers, other than of perfect squares, are irrational.

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Numerator

https://en.wiktionary.org/wiki/numerator

The number or expression written above the line in a fraction (such as 1 in ½).

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Order of Operations

https://en.wikipedia.org/wiki/Order_of_operations

The order of operations is a collection of rules that reflect conventions about which operations to perform first in order to evaluate a given mathematical expression.

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Perimeter

https://en.wikipedia.org/wiki/Perimeter

A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference.

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Product Rule of Exponents

Page 24, College Algebra 2e

For any real number $a$ and natural numbers $m$ and $n$, the product rule of exponents states that

$$a^m\cdot a^n = a^{m+n}$$

Quotient Rule of Exponents

Page 25, College Algebra 2e

For any real number $a$ and natural numbers $m$ and $n$, the quotient rule of exponents states that

$$\dfrac{a^m}{a^n} = a^{m-n}$$

Rational Number

https://commons.wikimedia.org/wiki/Category:Rational_numbers

A rational number is any number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, with the denominator $q$ not equal to zero. Since $q$ may be equal to 1, every integer is a rational number. This category represents all rational numbers, that is, those real numbers which can be represented in the form: a b $\frac{a}{b}$ ...where $a$ and $b$ are integers and $b$ is not equal to zero. All integers are rational, including zero.

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Real Number

https://en.wikipedia.org/wiki/Real_number

The real numbers include the rational numbers, such as the integer −5 and the fraction 4 / 3. The rest of the real numbers are called irrational numbers.

Real numbers can be thought of as all points on a line called the number line or real line, where the points corresponding to integers (..., −2, −1, 0, 1, 2, ...) are equally spaced.

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Repeating Decimal

https://en.wikipedia.org/wiki/Repeating_decimal

A repeating decimal or recurring decimal is a decimal representation of a number whose digits are periodic (repeating its values at regular intervals) and the infinitely repeated portion is not zero. It can be shown that a number is rational if and only if its decimal representation is repeating or terminating (i.e. all except finitely many digits are zero). For example, the decimal representation of $\frac{1}{3}$ becomes periodic just after the decimal point, repeating the single digit "3" forever, i.e. 0.333.... A more complicated example is $\frac{3227}{555}$, whose decimal becomes periodic at the second digit following the decimal point and then repeats the sequence "144" forever, i.e. 5.8144144144.... At present, there is no single universally accepted notation or phrasing for repeating decimals. Another example of this is $\frac{593}{53}$, which becomes periodic after the decimal point, repeating the 13-digit pattern "1886792452830" forever, i.e. 11.18867924528301886792452830....

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Surface Area

https://en.wikipedia.org/wiki/Surface_area

The surface area (symbol A) of a solid object is a measure of the total area that the surface of the object occupies.

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Terminating Decimal

https://en.wikipedia.org/wiki/Decimal

A decimal that only has a finite number of digits after the decimal seperator.

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note

• Content last updated on January 01, 2024
• College Algebra 2e is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0). Access for free at openstax.org. PDF link on Github
• Wikipedia text is licensed under Creative Commons Attribution-ShareAlike License 4.0 (CC BY-SA 4.0)
• Wiktionary text is licensed under Creative Commons Attribution-ShareAlike License (CC BY-SA 4.0)
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