Is 5.1 A Rational Number

Learning Objectives

In this department, you will:

• Classify a real number equally a natural, whole, integer, rational, or irrational number.
• Perform calculations using order of operations.
• Use the following properties of existent numbers: commutative, associative, distributive, inverse, and identity.
• Evaluate algebraic expressions.
• Simplify algebraic expressions.

It is often said that mathematics is the language of science. If this is true, then an essential part of the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate items. Farmers, cattle herders, and traders used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed length of cloth, for case. Doing so fabricated commerce possible, leading to improved communications and the spread of civilization.

Three to four one thousand years agone, Egyptians introduced fractions. They get-go used them to show reciprocals. Later, they used them to stand for the amount when a quantity was divided into equal parts.

Merely what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the beingness of nothing? From primeval times, people had thought of a "base of operations land" while counting and used various symbols to represent this zippo condition. However, information technology was non until nearly the fifth century CE in India that zippo was added to the number system and used as a numeral in calculations.

Clearly, there was also a need for numbers to represent loss or debt. In India, in the 7th century CE, negative numbers were used equally solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system fifty-fifty further.

Because of the development of the number system, nosotros tin now perform complex calculations using these and other categories of real numbers. In this section, nosotros volition explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions.

Classifying a Real Number

The numbers nosotros employ for counting, or enumerating items, are the natural numbers: ane, two, three, 4, v, then on. We describe them in prepare notation as $\left\{\mathrm{one},\mathrm{ii},3,\mathrm{...}\right\}$ where the ellipsis (…) indicates that the numbers go along to infinity. The natural numbers are, of course, also called the counting numbers. Any time nosotros enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, nosotros are using the gear up of natural numbers. The set of whole numbers is the set of natural numbers plus zero: $\left\{0,\mathrm{one},2,\mathrm{iii},\mathrm{...}\right\}.$

The set of integers adds the opposites of the natural numbers to the set of whole numbers: $\left\{\mathrm{...},\mathrm{-3},\mathrm{-2},\mathrm{-one},0,1,2,3,\mathrm{...}\right\}.$ It is useful to note that the set of integers is made upwards of three distinct subsets: negative integers, cipher, and positive integers. In this sense, the positive integers are just the natural numbers. Some other way to think about it is that the natural numbers are a subset of the integers.

$$\begin{array}{lllll}\stackrel{\text{negative integers}}{\stackrel{}{\dots ,\mathrm{-three},\mathrm{-2},\mathrm{-1},}}\hfill & \hfill & \stackrel{\text{zip}}{\stackrel{}{0,}}\hfill & \hfill & \stackrel{\text{positive integers}}{\stackrel{}{1,\mathrm{ii},3,\cdots }}\hfill \end{array}$$

The set up of rational numbers is written as $\left\{\frac{\mathrm{yard}}{\mathrm{due\; north}}|m\text{and}\mathrm{north}\text{are integers and}n\ne 0\right\}.$ Observe 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 tin too exist expressed in decimal form. Any rational number tin be represented as either:

1. ⓐa terminating decimal: $\frac{15}{8}=1.875,$ or
2. ⓑa repeating decimal: $\frac{4}{11}=0.36363636\dots =0.\overline{36}$

We utilize a line fatigued over the repeating cake of numbers instead of writing the group multiple times.

Case one

Writing Integers every bit Rational Numbers

Write each of the following as a rational number.

1. ⓐ7
2. ⓑ0
3. ⓒ–viii

Effort Information technology #1

Write each of the following as a rational number.

1. ⓐ11
2. ⓑthree
3. ⓒ–4

Example 2

Identifying Rational Numbers

Write each of the following rational numbers as either a terminating or repeating decimal.

1. ⓐ $-\frac{5}{7}$
2. ⓑ $\frac{15}{\mathrm{five}}$
3. ⓒ $\frac{13}{25}$

Try Information technology #2

Write each of the post-obit rational numbers as either a terminating or repeating decimal.

1. ⓐ $\frac{68}{17}$
2. ⓑ $\frac{8}{13}$
3. ⓒ $-\frac{17}{20}$

Irrational Numbers

At some point in the aboriginal by, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a foursquare with unit sides was not two or even $\frac{3}{\mathrm{two}},$ just was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a curl of cloth was a little bit more than than 3, just notwithstanding 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 exist expressed as a fraction of two integers. Information technology is impossible to describe this prepare of numbers by a single rule except to say that a number is irrational if information technology is not rational. Then we write this as shown.

$$\left\{h|h\text{is not a rational number}\right\}$$

Example iii

Differentiating Rational and Irrational Numbers

Determine whether each of the following numbers is rational or irrational. If it is rational, decide whether information technology is a terminating or repeating decimal.

1. ⓐ $\sqrt{25}$
2. ⓑ $\frac{33}{9}$
3. ⓒ $\sqrt{11}$
4. ⓓ $\frac{17}{34}$
5. ⓔ $0.3033033303333\dots$

Try It #3

Decide whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

1. ⓐ $\frac{\mathrm{vii}}{77}$
2. ⓑ $\sqrt{81}$
3. ⓒ $4.27027002700027\dots$
4. ⓓ $\frac{91}{\mathrm{thirteen}}$
5. ⓔ $\sqrt{39}$

Real Numbers

Given any number north, nosotros 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 exist divided into 3 subsets: negative real numbers, zero, and positive existent 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 tin can be visualized on a horizontal number line with an capricious point called 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 bones value) on either side of 0. Any real number corresponds to a unique position on the number line.The antipodal is also true: Each location on the number line corresponds to exactly 1 real number. This is known every bit a i-to-one correspondence. Nosotros refer to this as the existent number line as shown in Figure 1.

Effigy 1 The existent number line

Instance 4

Classifying Real Numbers

Allocate each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?

1. ⓐ $-\frac{10}{3}$
2. ⓑ $\sqrt{5}$
3. ⓒ $-\sqrt{289}$
4. ⓓ $\mathrm{-vi}\pi$
5. ⓔ $0.615384615384\dots$

Try It #4

Allocate each number equally either positive or negative and equally either rational or irrational. Does the number lie to the left or the right of 0 on the number line?

1. ⓐ $\sqrt{73}$
2. ⓑ $\mathrm{-11.411411411}\dots$
3. ⓒ $\frac{47}{19}$
4. ⓓ $-\frac{\sqrt{5}}{2}$
5. ⓔ $\mathrm{six.210735}$

Sets of Numbers as Subsets

Beginning with the natural numbers, we have expanded each ready to form a larger set up, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram, such every bit Figure 2.

Effigy 2 Sets of numbers
N: the fix of natural numbers
W: the set of whole numbers
I: the prepare of integers
Q: the set of rational numbers
Q´: the prepare of irrational numbers

Sets of Numbers

The set of natural numbers includes the numbers used for counting: $\left\{1,\mathrm{ii},3,\mathrm{...}\right\}.$

The fix of whole numbers is the prepare of natural numbers plus zero: $\left\{0,1,2,3,\mathrm{...}\right\}.$

The set up of integers adds the negative natural numbers to the set of whole numbers: $\left\{\mathrm{...},\mathrm{-3},\mathrm{-ii},\mathrm{-1},0,\mathrm{one},2,3,\mathrm{...}\right\}.$

The set up of rational numbers includes fractions written as $\left\{\frac{g}{n}|\mathrm{grand}\text{and}\mathrm{due\; north}\text{are integers and}\mathrm{due\; north}\ne 0\right\}.$

The prepare of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: $\left\{h|h\text{is non a rational number}\right\}.$

Instance 5

Differentiating the Sets of Numbers

Allocate each number every bit being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q′).

1. ⓐ $\sqrt{36}$
2. ⓑ $\frac{\mathrm{viii}}{3}$
3. ⓒ $\sqrt{73}$
4. ⓓ $\mathrm{-half\; dozen}$
5. ⓔ $\mathrm{three.2121121112}\dots$

Try It #5

Allocate each number as beingness a natural number (N), whole number (Westward), integer (I), rational number (Q), and/or irrational number (Q′).

1. ⓐ $-\frac{35}{7}$
2. ⓑ $0$
3. ⓒ $\sqrt{169}$
4. ⓓ $\sqrt{24}$
5. ⓔ $4.763763763\dots$

Performing Calculations Using the Lodge of Operations

When we multiply a number past itself, we square it or enhance information technology to a power of 2. For case, ${\mathrm{four}}^2=4\cdot \mathrm{four}=16.$ We tin raise any number to any power. In general, the exponential notation $a^n$ means that the number or variable $a$ is used as a factor $n$ times.

$$\begin{array}{l}{a}^n\hfill \end{array}=\stackrel{\mathrm{due\; north}\text{factors}}{\stackrel{}{a\cdot a\cdot a\cdot \dots \cdot a}}$$

In this notation, $a^n$ is read as the nthursday power of $a,$ or $a$ to the $\mathrm{north}$ where $a$ is chosen the base of operations and $n$ is called the exponent . A term in exponential notation may exist part of a mathematical expression, which is a combination of numbers and operations. For example, $24+6\cdot \frac{2}{3}-4^2$ is a mathematical expression.

To evaluate a mathematical expression, we perform the diverse operations. However, we do not perform them in whatsoever random social club. Nosotros use the lodge of operations. This is a sequence of rules for evaluating such expressions.

Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions then that annihilation appearing inside the symbols is treated as a unit of measurement. Additionally, fraction bars, radicals, and absolute value bars are treated equally group symbols. When evaluating a mathematical expression, brainstorm by simplifying expressions within grouping symbols.

The next step is to accost whatever exponents or radicals. Afterward, perform multiplication and division from left to correct and finally add-on and subtraction from left to correct.

Let'southward take a wait at the expression provided.

$$24+6\cdot \frac{2}{\mathrm{iii}}-4^2$$

There are no grouping symbols, and then we motility on to exponents or radicals. The number 4 is raised to a power of 2, and so simplify ${\mathrm{four}}^{\mathrm{two}}$ as 16.

$$\begin{array}{l}\hfill \\ \begin{array}{l}24+\mathrm{half-dozen}\cdot \frac{2}{3}-4^{\mathrm{ii}}\hfill \\ 24+6\cdot \frac{2}{3}-16\hfill \end{array}\hfill \end{array}$$

Next, perform multiplication or division, left to right.

$$\begin{array}{l}\hfill \\ \begin{array}{l}24+6\cdot \frac{2}{3}-16\hfill \\ 24+4-16\hfill \end{array}\hfill \end{array}$$

Lastly, perform addition or subtraction, left to correct.

$$\begin{array}{l}\hfill \\ 24+4-\mathrm{sixteen}\hfill \\ 28-16\hfill \\ 12\hfill \end{array}$$

Therefore, $24+\mathrm{six}\cdot \frac{\mathrm{two}}{\mathrm{three}}-4^2=12.$

For some complicated expressions, several passes through the social club of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the club of operations ensures that anyone simplifying the same mathematical expression will get the same result.

Order of Operations

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

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

How To

Given a mathematical expression, simplify information technology using the order of operations.

1. Footstep i. Simplify whatsoever expressions within group symbols.
2. Pace 2. Simplify whatsoever expressions containing exponents or radicals.
3. Step 3. Perform any multiplication and division in order, from left to right.
4. Step 4. Perform any add-on and subtraction in guild, from left to correct.

Case 6

Using the Order of Operations

Use the order of operations to evaluate each of the post-obit expressions.

1. ⓐ ${\left(3\cdot 2\right)}^2-4\left(\mathrm{vi}+2\right)$
2. ⓑ $\frac{5^{\mathrm{ii}}-4}{7}-\sqrt{11-\mathrm{ii}}$
3. ⓒ $6-\left|5-\mathrm{eight}\right|+\mathrm{iii}\left(\mathrm{four}-1\right)$
4. ⓓ $\frac{14-\mathrm{three}\cdot \mathrm{two}}{\mathrm{ii}\cdot \mathrm{v}-3^2}$
5. ⓔ $7\left(\mathrm{v}\cdot 3\right)-\mathrm{ii}\left[\left(6-3\right)-{\mathrm{four}}^{\mathrm{ii}}\right]+1$

Try It #half dozen

Utilise the guild of operations to evaluate each of the post-obit expressions.

1. ⓐ $\sqrt{{\mathrm{v}}^2-{\mathrm{four}}^{\mathrm{ii}}}+\mathrm{vii}{\left(\mathrm{v}-4\right)}^2$
2. ⓑ $1+\frac{7\cdot 5-8\cdot \mathrm{four}}{9-\mathrm{six}}$
3. ⓒ $|1.8-\mathrm{4.three}|+0.4\sqrt{15+10}$
4. ⓓ $\frac{\mathrm{one}}{2}\left[\mathrm{five}\cdot 3^2-7^2\right]+\frac{1}{3}\cdot 9^{\mathrm{ii}}$
5. ⓔ $\left[{\left(\mathrm{three}-\mathrm{viii}\right)}^2-4\right]-\left(3-8\right)$

Using Properties of Real Numbers

For some activities we perform, the society of certain operations does not matter, simply the gild of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does thing whether we put on shoes or socks first. The same thing is true for operations in mathematics.

Commutative Backdrop

The commutative property of add-on states that numbers may be added in whatsoever guild without affecting the sum.

$$a+b=b+a$$

We can ameliorate come across this human relationship when using real numbers.

$$\begin{array}{lllll}\left(\mathrm{-ii}\right)+7=5\hfill & \hfill & \text{and}\hfill & \hfill & 7+\left(\mathrm{-2}\right)=\mathrm{five}\hfill \end{array}$$

Similarly, the commutative belongings of multiplication states that numbers may be multiplied in whatsoever social club without affecting the product.

$$a\cdot b=b\cdot a$$

Once more, consider an example with real numbers.

$$\begin{array}{ccccc}\left(\mathrm{-eleven}\right)\cdot \left(\mathrm{-4}\right)=44& & \text{and}& & \left(\mathrm{-4}\right)\cdot \left(\mathrm{-11}\right)=44\end{array}$$

It is important to note that neither subtraction nor division is commutative. For example, $17-\mathrm{five}$ is not the same as $5-17.$ Similarly, $20÷\mathrm{v}\ne 5÷\mathrm{xx.}$

Associative Backdrop

The associative property of multiplication tells us that it does non affair how nosotros group numbers when multiplying. We can movement the grouping symbols to make the calculation easier, and the product remains the same.

$$a\left(bc\right)=\left(ab\right)c$$

Consider this instance.

$$\begin{array}{ccccc}\left(3\cdot 4\right)\cdot 5=\mathrm{lx}& & \text{and}& & 3\cdot \left(4\cdot 5\right)=60\end{array}$$

The associative holding of addition tells us that numbers may be grouped differently without affecting the sum.

$$a+\left(b+c\right)=\left(a+b\right)+c$$

This property tin be peculiarly helpful when dealing with negative integers. Consider this example.

$$\begin{array}{ccccc}\left[\mathrm{xv}+\left(\mathrm{-9}\right)\right]+23=29& & \text{and}& & 15+\left[\left(\mathrm{-9}\right)+23\right]=29\end{array}$$

Are subtraction and sectionalisation associative? Review these examples.

$$\begin{array}{cccccc}\hfill 8-(3-15)& \stackrel{?}{=}& (8-3)-\mathrm{fifteen}\hfill & \hfill 64÷(8÷4)& \stackrel{?}{=}& (64÷8)÷4\hfill \\ \hfill 8-(-12)& =& 5-15\hfill & \hfill 64÷2& \stackrel{?}{=}& \mathrm{eight}÷4\hfill \\ \hfill \mathrm{xx}& \ne & -10\hfill & \hfill 32& \ne & 2\hfill \end{array}$$

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

Distributive Belongings

The distributive property states that the product of a cistron times a sum is the sum of the factor times each term in the sum.

$$a\cdot \left(b+c\right)=a\cdot b+a\cdot c$$

This belongings combines both improver and multiplication (and is the only property to do and then). Let united states consider an case.

Note that iv is outside the grouping symbols, and so nosotros distribute the iv by multiplying information technology by 12, multiplying it by –7, and adding the products.

To exist more than precise when describing this holding, we say that multiplication distributes over addition. The reverse is not truthful, equally nosotros can see in this example.

$$\begin{array}{ccc}\hfill 6+(\mathrm{iii}\cdot \mathrm{five})& \stackrel{?}{=}& (\mathrm{half\; dozen}+3)\cdot (\mathrm{six}+5)\hfill \\ \hfill 6+(15)& \stackrel{?}{=}& (\mathrm{nine})\cdot (11)\hfill \\ \hfill 21& \ne & 99\hfill \end{array}$$

A special instance of the distributive property occurs when a sum of terms is subtracted.

$$a-b=a+\left(-b\right)$$

For example, consider the difference $12-\left(5+3\right).$ We can rewrite the departure of the two terms 12 and $\left(\mathrm{five}+3\right)$ by turning the subtraction expression into addition of the contrary. So instead of subtracting $\left(\mathrm{v}+\mathrm{three}\right),$ we add the contrary.

$$12+\left(\mathrm{-1}\right)\cdot \left(\mathrm{five}+3\right)$$

At present, distribute $\mathrm{-1}$ and simplify the issue.

$$\begin{array}{ccc}\hfill 12-(5+3)& =\hfill & 12+(\mathrm{-1})\cdot (5+3)\hfill \\ & =\hfill & 12+[(\mathrm{-i})\cdot 5+(\mathrm{-ane})\cdot 3]\hfill \\ & =\hfill & 12+(\mathrm{-eight})\hfill \\ & =\hfill & 4\hfill \end{array}$$

This seems like a lot of trouble for a elementary sum, but information technology illustrates a powerful upshot that volition 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 listen, we tin can rewrite the terminal example.

$$\begin{array}{ccc}\hfill 12-(5+\mathrm{iii})& =& 12+(\mathrm{-5}-3)\\ & =& 12+(\mathrm{-viii})\hfill \\ & =& \mathrm{iv}\hfill \end{array}$$

Identity Properties

The identity belongings of addition states that there is a unique number, chosen 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 at that place is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number.

$$a\cdot \mathrm{i}=a$$

For example, we have $\left(\mathrm{-vi}\right)+0=\mathrm{-6}$ and $23\cdot 1=23.$ At that place are no exceptions for these backdrop; they work for every real number, including 0 and 1.

Inverse Properties

The inverse property of add-on states that, for every real number a, in that location is a unique number, chosen the condiment inverse (or contrary), denoted by (−a), that, when added to the original number, results in the condiment identity, 0.

$$a+\left(-a\right)=0$$

For example, if $a=\mathrm{-8},$ the additive changed is viii, since $\left(\mathrm{-8}\right)+8=0.$

The inverse property of multiplication holds for all existent numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every existent 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, ane.

$$a\cdot \frac{1}{a}=1$$

For example, if $a=-\frac{2}{3},$ the reciprocal, denoted $\frac{\mathrm{i}}{a},$ is $-\frac{\mathrm{three}}{\mathrm{two}}$ considering

$$a\cdot \frac{\mathrm{i}}{a}=\left(-\frac{2}{3}\right)\cdot \left(-\frac{3}{2}\right)=1$$

Backdrop of Real Numbers

The post-obit properties hold for existent numbers a, b, and c.

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

Example 7

Using Backdrop of Existent Numbers

Use the properties of real numbers to rewrite and simplify each expression. Land which backdrop utilize.

1. ⓐ $3\cdot \mathrm{half\; dozen}+3\cdot 4$
2. ⓑ $\left(5+\mathrm{eight}\right)+\left(\mathrm{-8}\right)$
3. ⓒ $6-\left(15+9\right)$
4. ⓓ $\frac{\mathrm{four}}{\mathrm{seven}}\cdot \left(\frac{2}{\mathrm{iii}}\cdot \frac{\mathrm{vii}}{4}\right)$
5. ⓔ $100\cdot \left[0.75+\left(\mathrm{-2.38}\right)\right]$

Endeavor It #7

Use the backdrop of real numbers to rewrite and simplify each expression. Country which properties apply.

1. ⓐ $\left(-\frac{23}{5}\right)\cdot \left[\mathrm{xi}\cdot \left(-\frac{5}{23}\right)\right]$
2. ⓑ $5\cdot \left(\mathrm{vi.two}+0.4\right)$
3. ⓒ $18-\left(7\mathrm{-xv}\right)$
4. ⓓ $\frac{17}{18}+\left[\frac{\mathrm{four}}{\mathrm{ix}}+\left(-\frac{17}{18}\right)\right]$
5. ⓔ $6\cdot \left(\mathrm{-3}\right)+6\cdot 3$

Evaluating Algebraic Expressions

So far, the mathematical expressions nosotros accept seen have involved real numbers only. In mathematics, we may see expressions such as $x+\mathrm{v},\frac{4}{3}\pi r^3,$ or $\sqrt{\mathrm{two}{m}^3{n}^2}.$ In the expression $x+\mathrm{five},$ five is called a constant because information technology does non 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 drove of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.

We have already seen some existent number examples of exponential notation, a shorthand method of writing products of the aforementioned cistron. When variables are used, the constants and variables are treated the same style.

$$\begin{array}{cccccc}\hfill {(\mathrm{-3})}^{\mathrm{five}}& =& (\mathrm{-3})\cdot (\mathrm{-iii})\cdot (\mathrm{-three})\cdot (\mathrm{-three})\cdot (\mathrm{-iii})\hfill & \hfill x^{\mathrm{five}}& =& x\cdot \mathrm{ten}\cdot x\cdot \mathrm{10}\cdot x\hfill \\ \hfill {(\mathrm{two}\cdot 7)}^3& =& (2\cdot 7)\cdot (\mathrm{two}\cdot \mathrm{seven})\cdot (2\cdot 7)\hfill & \hfill {(yz)}^3& =& (yz)\cdot (yz)\cdot (yz)\hfill \end{array}$$

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

Any variable in an algebraic expression may take on or be assigned dissimilar values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to decide 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.

Example eight

Describing Algebraic Expressions

List the constants and variables for each algebraic expression.

1. ⓐ 10 + 5
2. ⓑ $\frac{\mathrm{four}}{\mathrm{iii}}\pi r^3$
3. ⓒ $\sqrt{2m^3{n}^2}$

Endeavor It #8

List the constants and variables for each algebraic expression.

1. ⓐ $2\pi r\left(r+h\right)$
2. ⓑ2(50 + W)
3. ⓒ $4y^{\mathrm{three}}+y$

Instance 9

Evaluating an Algebraic Expression at Different Values

Evaluate the expression $2x-\mathrm{seven}$ for each value for x.

1. ⓐ $x=0$
2. ⓑ $x=1$
3. ⓒ $x=\frac{\mathrm{one}}{2}$
4. ⓓ $\mathrm{ten}=\mathrm{-4}$

Effort It #9

Evaluate the expression $11-\mathrm{iii}y$ for each value for y.

1. ⓐ $y=2$
2. ⓑ $y=0$
3. ⓒ $y=\frac{\mathrm{ii}}{3}$
4. ⓓ $y=\mathrm{-5}$

Example 10

Evaluating Algebraic Expressions

Evaluate each expression for the given values.

1. ⓐ $\mathrm{ten}+5$ for $\mathrm{10}=\mathrm{-5}$
2. ⓑ $\frac{t}{2t\mathrm{-1}}$ for $t=\mathrm{ten}$
3. ⓒ $\frac{\mathrm{iv}}{3}\pi r^{\mathrm{three}}$ for $r=5$
4. ⓓ $a+ab+b$ for $a=11,b=\mathrm{-8}$
5. ⓔ $\sqrt{\mathrm{two}{m}^3{n}^2}$ for $m=2,\mathrm{due\; north}=3$

Try It #x

Evaluate each expression for the given values.

1. ⓐ $\frac{y+3}{y-\mathrm{iii}}$ for $y=5$
2. ⓑ $7-2t$ for $t=\mathrm{-2}$
3. ⓒ $\frac{1}{3}\pi r^2$ for $r=11$
4. ⓓ ${\left(p^{2}q\right)}^3$ for $p=\mathrm{-ii},q=3$
5. ⓔ $\mathrm{iv}\left(m-\mathrm{northward}\right)-5\left(\mathrm{north}-\mathrm{yard}\right)$ for $m=\frac{2}{3},n=\frac{1}{3}$

Formulas

An equation is a mathematical statement indicating that ii expressions are equal. The expressions tin can be numerical or algebraic. The equation is not inherently true or false, simply simply a proffer. The values that make the equation true, the solutions, are constitute using the backdrop of real numbers and other results. For example, the equation $2\mathrm{ten}+\mathrm{i}=7$ has the solution of 3 $$ because when nosotros substitute 3 for $\mathrm{ten}$ in the equation, we obtain the true statement $2\left(\mathrm{iii}\right)+1=7.$

A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a ways of finding the value of one quantity (often a single variable) in terms of another or other quantities. 1 of the near mutual 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 constitute by evaluating the expression $\pi r^{\mathrm{ii}}.$

Case 11

Using a Formula

A right circular cylinder with radius $r$ and height $h$ has the surface area $S$ (in square units) given by the formula $S=2\pi r\left(r+h\right).$ Encounter Effigy iii. Detect the surface expanse of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of $\pi .$

Effigy 3 Right round cylinder

Effort Information technology #eleven

A photograph with length 50 and width W is placed in a mat of width viii centimeters (cm). The area of the mat (in square centimeters, or cmⁱⁱ) is found to be $A=\left(L+16\right)\left(\mathrm{Westward}+16\right)-L\cdot W.$ See Figure 4. Detect the area of a mat for a photograph with length 32 cm and width 24 cm.

Effigy iv

Simplifying Algebraic Expressions

Sometimes we can simplify an algebraic expression to make information technology easier to evaluate or to utilize in another fashion. To practice and so, we employ the backdrop of real numbers. Nosotros tin can use the same properties in formulas because they contain algebraic expressions.

Example 12

Simplifying Algebraic Expressions

Simplify each algebraic expression.

1. ⓐ $3\mathrm{10}-2y+x-\mathrm{three}y-7$
2. ⓑ $2r-5\left(3-r\right)+\mathrm{iv}$
3. ⓒ $\left(4t-\frac{5}{4}s\right)-\left(\frac{2}{3}t+2s\right)$
4. ⓓ $\mathrm{ii}g\mathrm{north}-5g+\mathrm{iii}mn+n$

Endeavour Information technology #12

Simplify each algebraic expression.

1. ⓐ $\frac{2}{3}y-\mathrm{ii}\left(\frac{4}{\mathrm{iii}}y+z\right)$
2. ⓑ $\frac{5}{t}-2-\frac{\mathrm{three}}{t}+\mathrm{i}$
3. ⓒ $4p\left(q-1\right)+q\left(1-p\right)$
4. ⓓ $9r-\left(\mathrm{southward}+2r\right)+\left(6-s\right)$

Example thirteen

Simplifying a Formula

A rectangle with length $L$ and width $\mathrm{West}$ has a perimeter $P$ given past $P=L+W+L+W.$ Simplify this expression.

Endeavour It #xiii

If the amount $P$ is deposited into an account paying simple interest $r$ for time $t,$ the total value of the deposit $A$ is given past $A=P+Prt.$ Simplify the expression. (This formula will exist explored in more than particular later in the course.)

i.1 Department Exercises

Exact

one.

Is $\sqrt{\mathrm{two}}$ an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.

two .

What is the order of operations? What acronym is used to depict the order of operations, and what does it stand for?

3.

What practice the Associative Properties allow us to practice when following the gild of operations? Explain your answer.

Numeric

For the following exercises, simplify the given expression.

4 .

$10+2\times \left(5-\mathrm{iii}\right)$

5.

$\mathrm{half\; dozen}÷\mathrm{ii}-\left(81÷3^{\mathrm{two}}\right)$

6 .

$\mathrm{eighteen}+{\left(\mathrm{half-dozen}-\mathrm{viii}\right)}^3$

7.

$\mathrm{-two}\times {\left[\mathrm{sixteen}÷{\left(8-4\right)}^{\mathrm{two}}\right]}^2$

11.

$12÷\left(36÷9\right)+\mathrm{vi}$

12 .

${\left(4+\mathrm{v}\right)}^2÷3$

15.

$5+\left(\mathrm{half-dozen}+4\right)-\mathrm{xi}$

xviii .

$9-\left(3+11\right)\times 2$

20 .

$64÷\left(8+4\times 2\right)$

21.

$9+\mathrm{iv}\left(2^{\mathrm{ii}}\right)$

22 .

${\left(12÷3\times 3\right)}^{\mathrm{ii}}$

23.

$25÷5^2-7$

24 .

$\left(15-\mathrm{seven}\right)\times \left(\mathrm{iii}-7\right)$

25.

$\mathrm{two}\times 4-9\left(\mathrm{-1}\right)$

26 .

${\mathrm{four}}^2-25\times \frac{1}{5}$

27.

$12\left(3-\mathrm{ane}\right)÷\mathrm{half\; dozen}$

Algebraic

For the following exercises, evaluate the expression using the given value of the variable.

28 .

$8\left(\mathrm{10}+3\right)–64$ for $\mathrm{ten}=\mathrm{two}$

29.

$\mathrm{iv}y+8–2y$ for $y=3$

30 .

$\left(\mathrm{xi}a+3\right)-18a+4$ for $a=\mathrm{–2}$

31.

$\mathrm{iv}z-2z\left(\mathrm{ane}+4\right)–36$ for $z=\mathrm{v}$

32 .

$4y{\left(7-2\right)}^2+200$ for $y=\mathrm{–2}$

33.

$-{\left(\mathrm{ii}\mathrm{10}\right)}^{\mathrm{two}}+1+3$ for $x=2$

34 .

For the $\mathrm{eight}\left(2+4\right)-15b+b$ for $b=\mathrm{–iii}$

35.

$\mathrm{ii}\left(11c-4\right)–36$ for $c=0$

36 .

$\mathrm{iv}\left(3-\mathrm{i}\right)x–\mathrm{four}$ for $x=10$

37.

$\frac{1}{4}\left(\mathrm{eight}\mathrm{due\; west}-4^{\mathrm{two}}\right)$ for $w=1$

For the following exercises, simplify the expression.

38 .

$4x+x\left(13-7\right)$

39.

$2y-{\left(\mathrm{four}\right)}^{2}y-11$

40 .

$\frac{a}{2^3}\left(64\right)-12a÷6$

41.

$8b-\mathrm{four}b\left(3\right)+1$

42 .

$5\mathrm{fifty}÷3\mathrm{50}\times \left(9-\mathrm{six}\right)$

43.

$7z-3+z\times 6^2$

44 .

$4\times 3+18x÷\mathrm{nine}-12$

45.

$9\left(y+8\right)-27$

46 .

$\left(\frac{9}{6}t-4\right)2$

47.

$6+12b-3\times \mathrm{six}b$

48 .

$18y-2\left(1+\mathrm{vii}y\right)$

49.

${\left(\frac{4}{9}\right)}^2\times 27x$

50 .

$8\left(\mathrm{three}-g\right)+\mathrm{i}\left(-8\right)$

51.

$9\mathrm{10}+\mathrm{iv}x\left(\mathrm{two}+3\right)-4\left(2\mathrm{10}+3x\right)$

52 .

$5^2-4\left(3x\right)$

Real-World Applications

For the following exercises, consider this scenario: Fred earns $40 at the community garden. He spends $x on a streaming subscription, puts half of what is left in a savings business relationship, and gets another $five for walking his neighbor's dog.

53.

Write the expression that represents the number of dollars Fred keeps (and does non put in his savings account). Think the order of operations.

54 .

How much coin does Fred keep?

For the following exercises, solve the given problem.

55.

Co-ordinate to the U.South. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied past $\pi .$ Is the circumference of a quarter a whole number, a rational number, or an irrational number?

56 .

Jessica and her roommate, Adriana, have decided to share a alter jar for articulation expenses. Jessica put her loose change in the jar first, then Adriana put her change in the jar. Nosotros know that it does not matter in which guild the alter was added to the jar. What property of addition describes this fact?

For the post-obit exercises, consider this scenario: There is a mound of $\mathrm{thou}$ pounds of gravel in a quarry. Throughout the twenty-four hours, 400 pounds of gravel is added to the mound. 2 orders of 600 pounds are sold and the gravel is removed from the mound. At the terminate of the day, the mound has 1,200 pounds of gravel.

57.

Write the equation that describes the state of affairs.

For the following exercise, solve the given problem.

59.

Ramon runs the marketing section at their visitor. Their department gets a budget every year, and every year, they must spend the entire budget without going over. If they spend less than the budget, and so the section gets a smaller upkeep the following year. At the beginning of this year, Ramon got $2.5 million for the annual marketing upkeep. They must spend the upkeep such that $\mathrm{2,500,000}-x=0.$ What belongings of addition tells the states what the value of 10 must be?

Applied science

For the following exercises, use a graphing calculator to solve for x. Circular the answers to the nearest hundredth.

60 .

$\mathrm{0.v}{\left(12.3\right)}^{\mathrm{two}}-48x=\frac{3}{5}$

61.

${\left(0.25-0.75\right)}^2\mathrm{10}-\mathrm{vii.2}=9.9$

Extensions

62 .

If a whole number is not a natural number, what must the number be?

63.

Decide whether the statement is true or faux: The multiplicative inverse of a rational number is as well rational.

64 .

Make up one's mind whether the statement is true or faux: The product of a rational and irrational number is e'er irrational.

65.

Determine whether the simplified expression is rational or irrational: $\sqrt{\mathrm{-18}-4\left(5\right)\left(\mathrm{-i}\right)}.$

66 .

Make up one's mind whether the simplified expression is rational or irrational: $\sqrt{\mathrm{-16}+\mathrm{four}\left(\mathrm{five}\right)+5}.$

67.

The division of two natural numbers will always effect in what blazon of number?

68 .

What property of real numbers would simplify the post-obit expression: $4+7(\mathrm{10}-\mathrm{i})?$

Is 5.1 A Rational Number,

Source: https://openstax.org/books/college-algebra-2e/pages/1-1-real-numbers-algebra-essentials

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