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Real Numbers: Algebra Essentials

Classify a real number as a natural, whole, integer, rational, or irrational number.

Real Numbers: Algebra Essentials

Identify the study skills leading to success in a college level mathematics course.
Reflect on your past math experiences and create a plan for improvement.

It’s important to take the opportunity to reflect on your past experiences in math classes as you begin a new term. We can learn a lot from these reflections and thus work toward developing a strategy for improvement.
In the table below list 5 challenges you had in past math courses and list a possible solution that you could try this semester.

Challenge	Possible solution
1.
2.
3.
4.
5.

Write your math autobiography. Tell your math story by describing your past experiences as a learner of mathematics. Share how your attitudes have changed about math over the years if they have. Perhaps include what you love, hate, dread, appreciate, fear, look forward to, or find beauty in. This will help your teacher to better understand you and your current feelings about the discipline.
Share your autobiographies with your study group members. This helps to create a community in the classroom when common themes emerge.

Condensed — the full section is in OpenStax College Algebra 2e.

Classifying a Real Number

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 \(\{1,2,3,...\}\) where the ellipsis (…) 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. The set of whole numbers is the set of natural numbers plus zero: \(\{0,1,2,3,...\}.\)

The set of integers adds the opposites of the natural numbers to the set of whole numbers: \(\{...,-3,-2,-1,0,1,2,3,...\}.\) 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.

\[\begin{array}{lllll}\overset{\text{negative integers}}{\overset{}{\ldots ,-3,-2,-1,}} & & \overset{\text{zero}}{\overset{}{0,}} & & \overset{\text{positive integers}}{\overset{}{1,2,3,\cdots }}\end{array}\]

The set of rational numbers is written as \(\{\frac{m}{n}\ |m\ \text{and }n\ \text{are integers and }n\ne 0\}.\) 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 as a terminating or repeating decimal. Any rational number can be represented as either:

ⓐa terminating decimal: \(\frac{15}{8}=1.875,\) or
ⓑa repeating decimal: \(\frac{4}{11}=0.36363636\ldots =0.\overset{\bar}{36}\)

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

Example

Try it.

Show that each of the following is a rational number by writing it as a ratio of two integers.

ⓐ7
ⓑ0
ⓒ–8

Solution

Write a fraction with the integer in the numerator and 1 in the denominator.

ⓐ \(7=\frac{7}{1}\)
ⓑ \(0=\frac{0}{1}\)
ⓒ \(-8=-\frac{8}{1}\)

Condensed — the full section is in OpenStax College Algebra 2e.

Performing Calculations Using the Order of Operations

When we multiply a number by itself, we square it or raise it to a power of 2. For example, \({4}^{2}=4⋅4=16.\) We can 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.

\[\begin{array}{l}{a}^{n}\end{array}=\overset{n\ \text{factors}}{\overset{}{a⋅a⋅a⋅\ldots ⋅a}}\]

In this notation, \({a}^{n}\) is read as the nth power of \(a,\)or \(a\) to the \(n\) where \(a\) is called the base and \(n\) is called the exponent. A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operators, i.e., + addition, – subtraction, × multiplication, ÷ division. For example, \(24+6⋅\frac{2}{3}-{4}^{2}\) is a mathematical expression.

To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order 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 so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.

The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.

Let’s take a look at the expression provided.

\[24+6⋅\frac{2}{3}-{4}^{2}\]

There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify \({4}^{2}\) as 16.

\[\begin{array}{l} \\ \begin{array}{l}24+6⋅\frac{2}{3}-{4}^{2} \\ 24+6⋅\frac{2}{3}-16\end{array}\end{array}\]\[\begin{array}{l} \\ \begin{array}{l}24+6⋅\frac{2}{3}-16 \\ 24+4-16\end{array}\end{array}\]\[\begin{array}{l} \\ 24+4-16 \\ 28-16 \\ 12\end{array}\]

Condensed — the full section is in OpenStax College Algebra 2e.

Using Properties of Real Numbers

For some activities we perform, the order of certain operations does not matter, but the order 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 matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.

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 of 3 \(\) because when we substitute 3 for \(x\) in the equation, we obtain the true statement \(2(3)+1=7.\)

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 \(\pi {r}^{2}.\)

Example

Try it.

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

Solution

Evaluate the expression \(2\pi r(r+h)\) for \(r=6\) and \(h=9.\)

\[\begin{array}{lll}S & = & 2\pi r(r+h) \\ & = & 2\pi (6)[(6)+(9)] \\ & = & 2\pi (6)(15) \\ & = & 180\pi \end{array}\]

The surface area is \(180\pi\) square inches.

Condensed — the full section is in OpenStax College Algebra 2e.

Key Concepts

Rational numbers may be written as fractions or terminating or repeating decimals. See and .
Determine whether a number is rational or irrational by writing it as a decimal. See .
The rational numbers and irrational numbers make up the set of real numbers. See . A number can be classified as natural, whole, integer, rational, or irrational. See .
The order of operations is used to evaluate expressions. See .
The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of real numbers. These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties. See .
Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division. See . They take on a numerical value when evaluated by replacing variables with constants. See , , and
Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or evaluated as any mathematical expression. See and .

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Each of the behaviors or attitudes listed in the table above are associated with success in college mathematics. This means that students who use these strategies or are open to these beliefs are successful learners. Share your total score with your study group in class and be supportive of your fellow students!
Based on this survey create a list of the top 5 strategies that you currently utilize, and feel are most helpful to you.
Based on this survey create a list of the top 5 strategies that interest you, and that you feel could be most helpful to you this term. Plan on implementing these strategies.
Show that each of the following is a rational number by writing it as a ratio of two integers.
1. ⓐ7
2. ⓑ0
3. ⓒ–8
Mutasd meg a választ!
Write a fraction with the integer in the numerator and 1 in the denominator.
1. ⓐ \(7=\frac{7}{1}\)
2. ⓑ \(0=\frac{0}{1}\)
3. ⓒ \(-8=-\frac{8}{1}\)
Write each of the following as a rational number.
1. ⓐ11
2. ⓑ3
3. ⓒ–4
Mutasd meg a választ!
1. ⓐ \(\frac{11}{1}\)
2. ⓑ \(\frac{3}{1}\)
3. ⓒ \(-\frac{4}{1}\)
Write each of the following rational numbers as either a terminating or repeating decimal.
1. ⓐ \(-\frac{5}{7}\)
2. ⓑ \(\frac{15}{5}\)
3. ⓒ \(\frac{13}{25}\)
Mutasd meg a választ!
Write each fraction as a decimal by dividing the numerator by the denominator.
1. ⓐ \(-\frac{5}{7}=-0.\overset{\text{———}}{714285},\) a repeating decimal
2. ⓑ \(\frac{15}{5}=3\) (or 3.0), a terminating decimal
3. ⓒ \(\frac{13}{25}=0.52,\) a terminating decimal
Write each of the following rational numbers as either a terminating or repeating decimal.
1. ⓐ \(\frac{68}{17}\)
2. ⓑ \(\frac{8}{13}\)
3. ⓒ \(-\frac{17}{20}\)
Mutasd meg a választ!
1. ⓐ4 (or 4.0), terminating;
2. ⓑ \(0.\overset{\bar}{615384},\) repeating;
3. ⓒ–0.85, terminating
Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.
1. ⓐ \(\sqrt{25}\)
2. ⓑ \(\frac{33}{9}\)
3. ⓒ \(\sqrt{11}\)
4. ⓓ \(\frac{17}{34}\)
5. ⓔ \(0.3033033303333\ldots\)
Mutasd meg a választ!
1. ⓐ \(\sqrt{25}:\) This can be simplified as \(\sqrt{25}=5.\) Therefore, \(\sqrt{25}\) is rational.
2. ⓑ \(\frac{33}{9}:\) Because it is a fraction of integers, \(\frac{33}{9}\) is a rational number. Next, simplify and divide. \[\frac{33}{9}=\frac{\overset{11}{33}}{\underset{3}{9}}=\frac{11}{3}=3.\overset{\bar}{6}\]
  So, \(\frac{33}{9}\) is rational and a repeating decimal.
3. ⓒ \(\sqrt{11}:\sqrt{11}\) is irrational because 11 is not a perfect square and \(\sqrt{11}\) cannot be expressed as a fraction.
4. ⓓ \(\frac{17}{34}:\) Because it is a fraction of integers, \(\frac{17}{34}\) is a rational number. Simplify and divide. \[\frac{17}{34}=\frac{\overset{1}{17}}{\underset{2}{34}}=\frac{1}{2}=0.5\]
  So, \(\frac{17}{34}\) is rational and a terminating decimal.
5. ⓔ \(0.3033033303333\ldots\) is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.
Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.
1. ⓐ \(\frac{7}{77}\)
2. ⓑ \(\sqrt{81}\)
3. ⓒ \(4.27027002700027\ldots\)
4. ⓓ \(\frac{91}{13}\)
5. ⓔ \(\sqrt{39}\)
Mutasd meg a választ!
1. ⓐrational and repeating;
2. ⓑrational and terminating;
3. ⓒirrational;
4. ⓓrational and terminating;
5. ⓔirrational
Classify 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. ⓓ \(-6\pi\)
5. ⓔ \(0.615384615384\ldots\)
Mutasd meg a választ!
1. ⓐ \(-\frac{10}{3}\) is negative and rational. It lies to the left of 0 on the number line.
2. ⓑ \(\sqrt{5}\) is positive and irrational. It lies to the right of 0.
3. ⓒ \(-\sqrt{289}=-\sqrt{{17}^{2}}=-17\) is negative and rational. It lies to the left of 0.
4. ⓓ \(-6\pi\) is negative and irrational. It lies to the left of 0.
5. ⓔ \(0.615384615384\ldots\) is a repeating decimal so it is rational and positive. It lies to the right of 0.
Classify 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. ⓐ \(\sqrt{73}\)
2. ⓑ \(-11.411411411\ldots\)
3. ⓒ \(\frac{47}{19}\)
4. ⓓ \(-\frac{\sqrt{5}}{2}\)
5. ⓔ \(6.210735\)
Mutasd meg a választ!
1. ⓐpositive, irrational; right
2. ⓑnegative, rational; left
3. ⓒpositive, rational; right
4. ⓓnegative, irrational; left
5. ⓔpositive, rational; right
Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q′).
1. ⓐ \(\sqrt{36}\)
2. ⓑ \(\frac{8}{3}\)
3. ⓒ \(\sqrt{73}\)
4. ⓓ \(-6\)
5. ⓔ \(3.2121121112\ldots\)
Mutasd meg a választ!

N W I Q Q′
a. \(\sqrt{36}=6\) X X X X
b. \(\frac{8}{3}=2.\overset{\bar}{6}\) X
c. \(\sqrt{73}\) X
d. –6 X X
e. 3.2121121112... X
Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q′).
1. ⓐ \(-\frac{35}{7}\)
2. ⓑ \(0\)
3. ⓒ \(\sqrt{169}\)
4. ⓓ \(\sqrt{24}\)
5. ⓔ \(4.763763763\ldots\)
Mutasd meg a választ!

N W I Q Q'
a. \(-\frac{35}{7}\) X X
b. 0 X X X
c. \(\sqrt{169}\) X X X X
d. \(\sqrt{24}\) X
e. 4.763763763... X
Use the order of operations to evaluate each of the following expressions.
1. ⓐ \({(3⋅2)}^{2}-4(6+2)\)
2. ⓑ \(\frac{{5}^{2}-4}{7}-\sqrt{11-2}\)
3. ⓒ \(6-|5-8|+3(4-1)\)
4. ⓓ \(\frac{14-3⋅2}{2⋅5-{3}^{2}}\)
5. ⓔ \(7(5⋅3)-2[(6-3)-{4}^{2}]+1\)
Mutasd meg a választ!
1. ⓐ
  \(\begin{array}{llll}{(3⋅2)}^{2}-4(6+2) & = & {(6)}^{2}-4(8) & \ \text{Simplify parentheses.} \\ & = & 36-4(8) & \ \text{Simplify exponent.} \\ & = & 36-32 & \ \text{Simplify multiplication.} \\ & = & 4 & \ \text{Simplify subtraction.}\end{array}\)
2. ⓑ
  \(\begin{array}{llll}\frac{{5}^{2}-4}{7}-\sqrt{11-2} & = & \frac{{5}^{2}-4}{7}-\sqrt{9} & \ \text{Simplify grouping symbols (radical).} \\ & = & \frac{{5}^{2}-4}{7}-3 & \ \text{Simplify radical.} \\ & = & \frac{25-4}{7}-3 & \ \text{Simplify exponent.} \\ & = & \frac{21}{7}-3 & \ \text{Simplify subtraction in numerator.} \\ & = & 3-3 & \ \text{Simplify division.} \\ & = & 0 & \ \text{Simplify subtraction.}\end{array}\)
  Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.
3. ⓒ
  \(\begin{array}{llll}6-|5-8|+3|4-1| & = & 6-|-3|+3(3) & \ \text{Simplify inside grouping symbols.} \\ & = & 6-(3)+3(3) & \ \text{Simplify absolute value.} \\ & = & 6-3+9 & \ \text{Simplify multiplication.} \\ & = & 12 & \ \text{Simplify addition.}\end{array}\)
4. ⓓ
  \(\begin{array}{llll}\frac{14-3⋅2}{2⋅5-{3}^{2}} & = & \frac{14-3⋅2}{2⋅5-9} & \ \text{Simplify exponent.} \\ & = & \frac{14-6}{10-9} & \ \text{Simplify products.} \\ & = & \frac{8}{1} & \ \text{Simplify differences.} \\ & = & 8 & \ \text{Simplify quotient.}\end{array}\)
  In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.
5. ⓔ
  \(\begin{array}{llll}7(5⋅3)-2[(6-3)-{4}^{2}]+1 & = & 7(15)-2[(3)-{4}^{2}]+1 & \ \text{Simplify inside parentheses.} \\ & = & 7(15)-2(3-16)+1 & \ \text{Simplify exponent.} \\ & = & 7(15)-2(-13)+1 & \ \text{Subtract.} \\ & = & 105+26+1 & \ \text{Multiply.} \\ & = & 132 & \ \text{Add.}\end{array}\)
Use the order of operations to evaluate each of the following expressions.
1. ⓐ \(\sqrt{{5}^{2}-{4}^{2}}+7{(5-4)}^{2}\)
2. ⓑ \(1+\frac{7⋅5-8⋅4}{9-6}\)
3. ⓒ \(|1.8-4.3|+0.4\sqrt{15+10}\)
4. ⓓ \(\frac{1}{2}[5⋅{3}^{2}-{7}^{2}]+\frac{1}{3}⋅{9}^{2}\)
5. ⓔ \([{(3-8)}^{2}-4]-(3-8)\)
Mutasd meg a választ!
1. ⓐ10
2. ⓑ2
3. ⓒ4.5
4. ⓓ25
5. ⓔ26
Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.
1. ⓐ \(3⋅6+3⋅4\)
2. ⓑ \((5+8)+(-8)\)
3. ⓒ \(6-(15+9)\)
4. ⓓ \(\frac{4}{7}⋅(\frac{2}{3}⋅\frac{7}{4})\)
5. ⓔ \(100⋅[0.75+(-2.38)]\)
Mutasd meg a választ!
1. ⓐ
  \(\begin{array}{llll}3⋅6+3⋅4 & = & 3⋅(6+4) & \ \text{Distributive property.} \\ & = & 3⋅10 & \ \text{Simplify.} \\ & = & 30 & \ \text{Simplify.}\end{array}\)
2. ⓑ
  \(\begin{array}{llll}(5+8)+(-8) & = & 5+[8+(-8)] & \ \text{Associative property of addition.} \\ & = & 5+0 & \ \text{Inverse property of addition.} \\ & = & 5 & \ \text{Identity property of addition.}\end{array}\)
3. ⓒ
  \(\begin{array}{llll}6-(15+9) & = & 6+[(-15)+(-9)] & \ \text{Distributive property.} \\ & = & 6+(-24) & \ \text{Simplify.} \\ & = & -18 & \ \text{Simplify.}\end{array}\)
4. ⓓ
  \(\begin{array}{llll}\frac{4}{7}⋅(\frac{2}{3}⋅\frac{7}{4}) & = & \frac{4}{7}⋅(\frac{7}{4}⋅\frac{2}{3}) & \ \text{Commutative property of multiplication.} \\ & = & (\frac{4}{7}⋅\frac{7}{4})⋅\frac{2}{3} & \ \text{Associative property of multiplication.} \\ & = & 1⋅\frac{2}{3} & \ \text{Inverse property of multiplication.} \\ & = & \frac{2}{3} & \ \text{Identity property of multiplication.}\end{array}\)
5. ⓔ
  \(\begin{array}{llll}100⋅[0.75+(-2.38)] & = & 100⋅0.75+100⋅(-2.38) & \ \text{Distributive property.} \\ & = & 75+(-238) & \ \text{Simplify.} \\ & = & -163 & \ \text{Simplify.}\end{array}\)
Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.
1. ⓐ \((-\frac{23}{5})⋅[11⋅(-\frac{5}{23})]\)
2. ⓑ \(5⋅(6.2+0.4)\)
3. ⓒ \(18-(7-15)\)
4. ⓓ \(\frac{17}{18}+[\frac{4}{9}+(-\frac{17}{18})]\)
5. ⓔ \(6⋅(-3)+6⋅3\)
Mutasd meg a választ!
1. ⓐ11, commutative property of multiplication, associative property of multiplication, inverse property of multiplication, identity property of multiplication;
2. ⓑ33, distributive property;
3. ⓒ26, distributive property;
4. ⓓ \(\frac{4}{9},\) commutative property of addition, associative property of addition, inverse property of addition, identity property of addition;
5. ⓔ0, distributive property, inverse property of addition
List the constants and variables for each algebraic expression.
1. ⓐ x + 5
2. ⓑ \(\frac{4}{3}\pi {r}^{3}\)
3. ⓒ \(\sqrt{2{m}^{3}{n}^{2}}\)
Mutasd meg a választ!

Constants Variables
a. x + 5 5 x
b. \(\frac{4}{3}\pi {r}^{3}\) \(\frac{4}{3},\pi\) \(r\)
c. \(\sqrt{2{m}^{3}{n}^{2}}\) 2 \(m,n\)
List the constants and variables for each algebraic expression.
1. ⓐ \(2\pi r(r+h)\)
2. ⓑ2(L + W)
3. ⓒ \(4{y}^{3}+y\)
Mutasd meg a választ!

Constants Variables
a. \(2\pi r(r+h)\) \(2,\pi\) \(r,h\)
b. 2(L + W) 2 L, W
c. \(4{y}^{3}+y\) 4 \(y\)
Evaluate the expression \(2x-7\) for each value for x.
1. ⓐ \(x=0\)
2. ⓑ \(x=1\)
3. ⓒ \(x=\frac{1}{2}\)
4. ⓓ \(x=-4\)
Mutasd meg a választ!
1. ⓐ Substitute 0 for \(x.\) \[\begin{array}{lll}2x-7 & = & 2(0)-7 \\ & = & 0-7 \\ & = & -7\end{array}\]
2. ⓑSubstitute 1 for \(x.\) \[\begin{array}{lll}2x-7 & = & 2(1)-7 \\ & = & 2-7 \\ & = & -5\end{array}\]
3. ⓒ Substitute \(\frac{1}{2}\) for \(x.\) \[\begin{array}{lll}2x-7 & = & 2(\frac{1}{2})-7 \\ & = & 1-7 \\ & = & -6\end{array}\]
4. ⓓSubstitute \(-4\) for \(x.\) \[\begin{array}{lll}2x-7 & = & 2(-4)-7 \\ & = & -8-7 \\ & = & -15\end{array}\]
Evaluate the expression \(11-3y\) for each value for y.
1. ⓐ \(y=2\)
2. ⓑ \(y=0\)
3. ⓒ \(y=\frac{2}{3}\)
4. ⓓ \(y=-5\)
Mutasd meg a választ!
1. ⓐ5;
2. ⓑ11;
3. ⓒ9;
4. ⓓ26
Evaluate each expression for the given values.
1. ⓐ \(x+5\) for \(x=-5\)
2. ⓑ \(\frac{t}{2t-1}\) for \(t=10\)
3. ⓒ \(\frac{4}{3}\pi {r}^{3}\) for \(r=5\)
4. ⓓ \(a+ab+b\) for \(a=11,b=-8\)
5. ⓔ \(\sqrt{2{m}^{3}{n}^{2}}\) for \(m=2,n=3\)
Mutasd meg a választ!
1. ⓐ Substitute \(-5\) for \(x.\) \[\begin{array}{lll}x+5 & = & (-5)+5 \\ & = & 0\end{array}\]
2. ⓑ Substitute 10 for \(t.\) \[\begin{array}{lll}\frac{t}{2t-1} & = & \frac{(10)}{2(10)-1} \\ & = & \frac{10}{20-1} \\ & = & \frac{10}{19}\end{array}\]
3. ⓒ Substitute 5 for \(r.\) \[\begin{array}{lll}\frac{4}{3}\pi {r}^{3} & = & \frac{4}{3}\pi {(5)}^{3} \\ & = & \frac{4}{3}\pi (125) \\ & = & \frac{500}{3}\pi \end{array}\]
4. ⓓ Substitute 11 for \(a\) and –8 for \(b.\) \[\begin{array}{lll}a+ab+b & = & (11)+(11)(-8)+(-8) \\ & = & 11-88-8 \\ & = & -85\end{array}\]
5. ⓔ Substitute 2 for \(m\) and 3 for \(n.\) \[\begin{array}{lll}\sqrt{2{m}^{3}{n}^{2}} & = & \sqrt{2{(2)}^{3}{(3)}^{2}} \\ & = & \sqrt{2(8)(9)} \\ & = & \sqrt{144} \\ & = & 12\end{array}\]
Evaluate each expression for the given values.
1. ⓐ \(\frac{y+3}{y-3}\) for \(y=5\)
2. ⓑ \(7-2t\) for \(t=-2\)
3. ⓒ \(\frac{1}{3}\pi {r}^{2}\) for \(r=11\)
4. ⓓ \({({p}^{2}q)}^{3}\) for \(p=-2,q=3\)
5. ⓔ \(4(m-n)-5(n-m)\) for \(m=\frac{2}{3},n=\frac{1}{3}\)
Mutasd meg a választ!
1. ⓐ4;
2. ⓑ11;
3. ⓒ \(\frac{121}{3}\pi\) ;
4. ⓓ1728;
5. ⓔ3
A right circular cylinder with radius \(r\) and height \(h\) has the surface area \(S\) (in square units) given by the formula \(S=2\pi r(r+h).\) See . Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of \(\pi .\)

Mutasd meg a választ!

Evaluate the expression \(2\pi r(r+h)\) for \(r=6\) and \(h=9.\)
\[\begin{array}{lll}S & = & 2\pi r(r+h) \\ & = & 2\pi (6)[(6)+(9)] \\ & = & 2\pi (6)(15) \\ & = & 180\pi \end{array}\]
The surface area is \(180\pi\) square inches.
A photograph with length L and width W is placed in a mat of width 8 centimeters (cm). The area of the mat (in square centimeters, or cm²) is found to be \(A=(L+16)(W+16)-L⋅W.\) See . Find the area of a mat for a photograph with length 32 cm and width 24 cm.

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1,152 cm²
Simplify each algebraic expression.
1. ⓐ \(3x-2y+x-3y-7\)
2. ⓑ \(2r-5(3-r)+4\)
3. ⓒ \((4t-\frac{5}{4}s)-(\frac{2}{3}t+2s)\)
4. ⓓ \(2mn-5m+3mn+n\)
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1. ⓐ
  \(\begin{array}{llll}3x-2y+x-3y-7 & = & 3x+x-2y-3y-7 & \ \text{Commutative property of addition.} \\ & = & 4x-5y-7 & \ \text{Simplify.}\end{array}\)
2. ⓑ
  \(\begin{array}{llll}2r-5(3-r)+4 & = & 2r-15+5r+4 & \ \text{Distributive property.} \\ & = & 2r+5r-15+4 & \ \text{Commutative property of addition.} \\ & = & 7r-11 & \ \text{Simplify.}\end{array}\)
3. ⓒ
  \(\begin{array}{llll}(4t-\frac{5}{4}s)-(\frac{2}{3}t+2s) & = & 4t-\frac{5}{4}s-\frac{2}{3}t-2s & \ \text{Distributive property.} \\ & = & 4t-\frac{2}{3}t-\frac{5}{4}s-2s & \ \text{Commutative property of addition.} \\ & = & \frac{10}{3}t-\frac{13}{4}s & \ \text{Simplify.}\end{array}\)
4. ⓓ
  \(\begin{array}{llll}2mn-5m+3mn+n & = & 2mn+3mn-5m+n & \ \text{Commutative property of addition.} \\ & = & \ 5mn-5m+n & \ \text{Simplify.}\end{array}\)
Simplify each algebraic expression.
1. ⓐ \(\frac{2}{3}y-2(\frac{4}{3}y+z)\)
2. ⓑ \(\frac{5}{t}-2-\frac{3}{t}+1\)
3. ⓒ \(4p(q-1)+q(1-p)\)
4. ⓓ \(9r-(s+2r)+(6-s)\)
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1. ⓐ \(-2y-2z\ \text{or }-2(y+z);\)
2. ⓑ \(\frac{2}{t}-1;\)
3. ⓒ \(3pq-4p+q;\)
4. ⓓ \(7r-2s+6\)
A rectangle with length \(L\) and width \(W\) has a perimeter \(P\) given by \(P=L+W+L+W.\) Simplify this expression.

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\[\begin{array}{llll}P & = & L+W+L+W & \\ P & = & L+L+W+W & \ \text{Commutative property of addition} \\ P & = & 2L+2W & \ \text{Simplify} \\ P & = & 2(L+W) & \ \text{Distributive property}\end{array}\]
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 by \(A=P+Prt.\) Simplify the expression. (This formula will be explored in more detail later in the course.)

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\(A=P(1+rt)\)
Is \(\sqrt{2}\) an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.

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irrational number. The square root of two does not terminate, and it does not repeat a pattern. It cannot be written as a quotient of two integers, so it is irrational.
What is the order of operations? What acronym is used to describe the order of operations, and what does it stand for?
What do the Associative Properties allow us to do when following the order of operations? Explain your answer.

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The Associative Properties state that the sum or product of multiple numbers can be grouped differently without affecting the result. This is because the same operation is performed (either addition or subtraction), so the terms can be re-ordered.
\(10+2\ \times \ (5-3)\)
\(6\div 2-(81\div {3}^{2})\)

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\(-6\)
\(18+{(6-8)}^{3}\)
\(-2\ \times \ {[16\div {(8-4)}^{2}]}^{2}\)

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\(-2\)
\(4-6+2\ \times \ 7\)
\(3(5-8)\)

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\(-9\)
\(4+6-10\div 2\)
\(12\div (36\div 9)+6\)

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9

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

imaginary unit
i² = −1.

\neq

not equal
The two sides are different.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Real Numbers: Algebra Essentials

Classify a real number as a natural, whole, integer, rational, or irrational number.
Perform calculations using order of operations.
Use the following properties of real numbers: commutative, associative, distributive, inverse, and identity.
Evaluate algebraic expressions.
Simplify algebraic expressions.
Identify the study skills leading to success in a college level mathematics course.
Reflect on your past math experiences and create a plan for improvement.

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Próbáld a sajátodat.

Parts of this page are adapted from OpenStax College Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Még több Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value

	N	W	I	Q	Q′
a. \(\sqrt{36}=6\)	X	X	X	X
b. \(\frac{8}{3}=2.\overset{\bar}{6}\)				X
c. \(\sqrt{73}\)					X
d. –6			X	X
e. 3.2121121112...					X

	N	W	I	Q	Q'
a. \(-\frac{35}{7}\)			X	X
b. 0		X	X	X
c. \(\sqrt{169}\)	X	X	X	X
d. \(\sqrt{24}\)					X
e. 4.763763763...				X

	Constants	Variables
a. x + 5	5	x
b. \(\frac{4}{3}\pi {r}^{3}\)	\(\frac{4}{3},\pi\)	\(r\)
c. \(\sqrt{2{m}^{3}{n}^{2}}\)	2	\(m,n\)

	Constants	Variables
a. \(2\pi r(r+h)\)	\(2,\pi\)	\(r,h\)
b. 2(L + W)	2	L, W
c. \(4{y}^{3}+y\)	4	\(y\)