MTH120 College Algebra Chapter 1.1

1.1 Real Numbers: Algebra Essentials 

1.1 Real Numbers: Algebra Essentials

Since chapters in Algebra are pretty long and contain a lot of data they will be separated into parts to make it easier to go through. 

Real numbers can be classified into several categories based on their properties and characteristics:

1. Natural Numbers (N): Natural numbers are positive whole numbers used for counting and ordering. They start from 1 and go on indefinitely (1, 2, 3, 4, ...).

2. Whole Numbers (W): Whole numbers include all natural numbers and zero (0, 1, 2, 3, 4, ...).

3. Integers (Z): Integers include all positive and negative whole numbers, along with zero (..., -3, -2, -1, 0, 1, 2, 3, ...).

4. Rational Numbers (Q): Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. These include fractions, integers, and whole numbers. For example, 1/2, -3, and 7/1 are rational numbers.

5. Irrational Numbers: Irrational numbers are real numbers that cannot be expressed as a fraction of two integers. They have non-repeating, non-terminating decimal expansions. Examples include the square root of 2 (√2), pi (π), and the mathematical constant "e."

To classify a specific real number, you would need to examine its properties and determine which category it belongs to based on the definitions provided above. Here are a few examples:

• 5: This is a natural number, a whole number, and an integer.

• -3: This is an integer and a whole number.

• 1/2: This is a rational number because it can be expressed as a fraction of two integers (1 and 2).

• √2: This is an irrational number because its decimal representation is non-repeating and non-terminating.

• π (pi): This is an irrational number because it cannot be expressed as a fraction of two integers, and its decimal representation is non-repeating and non-terminating.

Use the following properties of real numbers: commutative, associative, distributive, inverse, and identity.

These examples demonstrate how the properties of real numbers can be applied to simplify expressions, solve equations, and manipulate numbers in various ways.

1. Commutative Property of Addition: This property states that the order of numbers in addition doesn't affect the result.

   Example: $�+�=�+�$

   Let's use this property with real numbers:

   $3+5=5+3=8$

2. Associative Property of Multiplication: This property states that the grouping of numbers in multiplication doesn't affect the result.

   Example: $�\cdot (�\cdot �)=(�\cdot �)\cdot �$

   Using this property with real numbers:

   $2\cdot (3\cdot 4)=(2\cdot 3)\cdot 4=24$

3. Distributive Property: This property relates multiplication and addition, allowing us to distribute a factor to each term inside parentheses.

   Example: $�\cdot (�+�)=(�\cdot �)+(�\cdot �)$

   Applying the distributive property to real numbers:

   $2\cdot (3+4)=(2\cdot 3)+(2\cdot 4)=14$

4. Inverse Property of Addition: This property states that for every real number $�$, there exists an additive inverse $-�$ such that $�+(-�)=0$.

   Example: If $�=7$, then its additive inverse is $-7$, and $7+(-7)=0$.

5. Identity Property of Multiplication: This property states that for any real number $�$, $�\cdot 1=�$.

   Example: $5\cdot 1=5$

Now, let's combine some of these properties:

6. Using Commutative and Associative Properties Together:

   $(2+3)+(4+5)=(3+2)+(5+4)=5+9=14$

7. Using Distributive Property to Simplify:

   $3\cdot (4+2)=3\cdot 4+3\cdot 2=12+6=18$

8. Using Inverse Property to Solve Equations:

   $�+5=10$ To find the value of $�$, we can subtract 5 from both sides using the inverse property of addition: $�+5-5=10-5$ $�=5$

Expression 1: Simplify $2�+3�$.

To simplify this expression, combine like terms:

$2�+3�=5�$

So, $2�+3�$ simplifies to $5�$.

Expression 2: Simplify $\frac{4�}{2}$.

To simplify this expression, perform the division:

$\frac{4�}{2}=2�$

So, $\frac{4�}{2}$ simplifies to $2�$.

Expression 3: Simplify $3(�+4)-2(2�-1)$.

To simplify this expression, apply the distributive property and then combine like terms:

$3(�+4)-2(2�-1)=3�+12-4�+2$

Now, combine like terms:

$3�-4�+12+2=-�+14$

So, $3(�+4)-2(2�-1)$ simplifies to $-�+14$.

Expression 4: Simplify $2(�-3)+3(2�+1)$.

Again, apply the distributive property and then combine like terms:

$2(�-3)+3(2�+1)=2�-6+6�+3$

Combine like terms:

$2�+6�-6+3=8�-3$

So, $2(�-3)+3(2�+1)$ simplifies to $8�-3$.

To classify a real number into one of the common categories (natural, whole, integer, rational, or irrational), you need to understand the characteristics of each category and then examine the given number to determine where it belongs. Here's a step-by-step guide on how to classify a real number:

1. Natural Numbers (N): Natural numbers are positive counting numbers, starting from 1 and going on infinitely (1, 2, 3, 4, ...). They do not include zero or negative numbers.

2. Whole Numbers (W): Whole numbers include all the natural numbers and zero (0, 1, 2, 3, 4, ...).

3. Integers (Z): Integers include all positive and negative whole numbers along with zero (..., -3, -2, -1, 0, 1, 2, 3, ...).

4. Rational Numbers (Q): Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. These include fractions, integers, and whole numbers. For example, 1/2, -3, and 7/1 are rational numbers.

5. Irrational Numbers: Irrational numbers are real numbers that cannot be expressed as a fraction of two integers. They have non-repeating, non-terminating decimal expansions. Examples include the square root of 2 (√2), pi (π), and the mathematical constant "e."

To classify a specific real number, follow these steps:

1. Identify the number you want to classify (e.g., 5, -2, 3/4, √9, π).

2. Determine if the number is positive or negative.

3. Determine if the number is a fraction or a whole number (an integer, natural number, or zero).

4. If it's a fraction, check if it can be expressed as the quotient of two integers. If yes, it's a rational number. If not, it's irrational.

Let's classify a few examples:

• Example 1: Classify -5.

  • It's a negative whole number, so it's an integer.

• Example 2: Classify 0.

  • It's a whole number, but it's also an integer.

• Example 3: Classify 1/3.

  • It's a fraction, and it can't be expressed as the quotient of two integers. So, it's irrational.

• Example 4: Classify √25.

  • It's a positive whole number, so it's an integer.

• Example 5: Classify 3.14 (an approximation of π).

  • It's a positive decimal number that cannot be expressed as a fraction of two integers. So, it's irrational.

These examples illustrate how to classify real numbers based on their properties and characteristics.

Writing Integers as Rational Numbers:

Write each of the following as a rational number. 5 0 –9

To write each of the given integers (5, 0, -9) as a rational number, you can express them with a denominator of 1, which effectively turns them into fractions. In this form, any integer is a rational number because it's expressed as the quotient of an integer (the numerator) and 1 (the denominator). Here are the rational representations:

1. 5 as a Rational Number:

   • $5=\frac{5}{1}$

2. 0 as a Rational Number:

   • $0=\frac{0}{1}$
   • Note that any number divided by 1 is itself, so 0 divided by 1 is still 0.

3. -9 as a Rational Number:

   • $-9=\frac{-9}{1}$

So, all three of these integers (5, 0, -9) can be expressed as rational numbers by simply placing them over a denominator of 1.

Write each of the following as a rational number. 14 2 –1

To write each of the given integers (14, 2, -1) as rational numbers, you can express them with a denominator of 1, which effectively turns them into fractions. In this form, any integer is a rational number because it's expressed as the quotient of an integer (the numerator) and 1 (the denominator). Here are the rational representations:

1. 14 as a Rational Number:

   • $14=\frac{14}{1}$

2. 2 as a Rational Number:

   • $2=\frac{2}{1}$

3. -1 as a Rational Number:

   • $-1=\frac{-1}{1}$

So, all three of these integers (14, 2, -1) can be expressed as rational numbers by simply placing them over a denominator of 1.

Identifying Rational Numbers:

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

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

1. −57$-\frac{5}{\mathrm{7<span\; style="font-family:\; "Neue\; Helvetica\; W01",\; Helvetica,\; Arial,\; sans-serif;\; font-size:\; 16px;\; word-spacing:\; normal;"> </span>}}$
2. 155$\frac{15}{5}$
3. 1325
2. −57=−0.714285———,$-\frac{5}{7}=\mathrm{-0.}\stackrel{\text{———}}{714285},$ a repeating decimal

155=3$\frac{15}{5}=3$ (or 3.0), a terminating decimal

1325=0.52,$\frac{13}{25}=0.52,$ a terminating decimal

Irrational numbers are real numbers that cannot be expressed as the quotient or fraction of two integers, and they have non-repeating, non-terminating decimal expansions. In other words, they are numbers whose decimal representations go on forever without repeating any specific pattern.

Here are some well-known examples of irrational numbers:

1. π (Pi): Pi is the ratio of the circumference of a circle to its diameter. Its decimal representation starts with 3.14159 and goes on indefinitely without repeating.

2. √2 (Square Root of 2): The square root of 2 is an irrational number because it cannot be expressed as a fraction of two integers. Its decimal representation is approximately 1.41421356... and continues infinitely without repeating.

3. e (Euler's Number): Euler's number is another well-known irrational number. Its decimal representation begins with 2.71828 and goes on indefinitely without repeating.

4. √3 (Square Root of 3): The square root of 3 is irrational, and its decimal representation is approximately 1.732050807... with a non-repeating and non-terminating pattern.

5. √5 (Square Root of 5): Like other square roots of non-perfect squares, the square root of 5 is irrational. Its decimal representation is approximately 2.236067977... and continues infinitely.

These are just a few examples of irrational numbers. There are infinitely many irrational numbers, and they play a crucial role in mathematics and various scientific fields. They are essential for solving certain mathematical problems and are used in various mathematical and scientific calculations.

Here are a couple of examples of problems involving irrational numbers along with their solutions:

Problem 1: Calculate the exact value of $\sqrt{18}$ without using a calculator.

Solution: To find the exact value of $\sqrt{18}$, you can simplify it by factoring out the largest perfect square from under the square root sign:

$\sqrt{18}=\sqrt{9\cdot 2}$

Now, take the square root of the perfect square (9) and leave the square root of 2 under the radical sign:

$\sqrt{9}\cdot \sqrt{2}=3\sqrt{2}$

So, $\sqrt{18}$ is equal to $3\sqrt{2}$.

Problem 2: Calculate the exact value of $\sin \left(\frac{�}{4}\right)$ without using a calculator.

Solution: To find the sine of $\frac{�}{4}$, you can use the fact that $\sin \left(\frac{�}{4}\right)=\frac{\sqrt{2}}{2}$. This is based on the unit circle and trigonometric identities. So, the exact value of $\sin \left(\frac{�}{4}\right)$ is $\frac{\sqrt{2}}{2}$.

Both of these problems involve irrational numbers because $\sqrt{18}$ and $\frac{\sqrt{2}}{2}$ cannot be expressed as simple fractions or terminating decimals. They are exact representations of these irrational values.

Problem 3: Find the exact value of $\sin \left(\frac{3�}{4}\right)$ without using a calculator.

Solution: To find the sine of $\frac{3�}{4}$, you can use the fact that $\sin \left(\frac{3�}{4}\right)=\frac{\sqrt{2}}{2}$. This value is based on the unit circle and trigonometric identities, similar to the previous problem. So, the exact value of $\sin \left(\frac{3�}{4}\right)$ is $\frac{\sqrt{2}}{2}$.

Problem 4: Calculate the exact value of $\sqrt{50}$ without using a calculator.

Solution: To find the exact value of $\sqrt{50}$, you can simplify it by factoring out the largest perfect square from under the square root sign:

$\sqrt{50}=\sqrt{25\cdot 2}$

Now, take the square root of the perfect square (25) and leave the square root of 2 under the radical sign:

$\sqrt{25}\cdot \sqrt{2}=5\sqrt{2}$

So, $\sqrt{50}$ is equal to $5\sqrt{2}$.

Both of these problems involve more advanced calculations with irrational numbers, requiring knowledge of trigonometric identities and simplification techniques.

Real numbers are a comprehensive set of numbers that includes all rational and irrational numbers. They form the basis of the number system used in mathematics and can be represented on the real number line. Real numbers are used to describe quantities, measurements, and values in various contexts, including mathematics, science, engineering, and everyday life.

Here are some key characteristics and categories of real numbers:

1. Rational Numbers (Q): Rational numbers are real numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. Examples include fractions (e.g., 1/2), integers (e.g., -3), and whole numbers (e.g., 5).

2. Irrational Numbers: Irrational numbers are real numbers that cannot be expressed as the quotient of two integers. They have non-repeating, non-terminating decimal representations. Examples include the square root of 2 (√2), pi (π), and the golden ratio (φ).

3. Natural Numbers (N): Natural numbers are positive whole numbers used for counting and ordering. They start from 1 and continue indefinitely (1, 2, 3, 4, ...).

4. Whole Numbers (W): Whole numbers include all natural numbers and zero (0, 1, 2, 3, 4, ...).

5. Integers (Z): Integers include all positive and negative whole numbers along with zero (..., -3, -2, -1, 0, 1, 2, 3, ...).

6. Positive and Negative Numbers: Real numbers can be classified as positive (greater than zero), negative (less than zero), or zero.

7. Decimal Numbers: Real numbers can be represented as decimal fractions, which can be either terminating (ending) or non-terminating (repeating or non-repeating).

8. Infinite and Finite Decimals: Some real numbers have infinite decimal expansions (e.g., π), while others have finite decimal representations (e.g., 0.25).

9. Algebraic and Transcendental Numbers: Real numbers can also be classified as algebraic (solutions of polynomial equations with integer coefficients) or transcendental (not algebraic). Most irrational numbers are transcendental.

Here are some questions that involve classifying real numbers into various categories, along with their answers:

Question 1: Classify each of the following numbers as natural, whole, integer, rational, or irrational:

a) 17 b) -3 c) 4/5 d) √16 e) π (pi)

Answer 1: a) 17 is a natural number, a whole number, an integer, and a rational number (can be expressed as 17/1). b) -3 is an integer and a rational number (can be expressed as -3/1). c) 4/5 is a rational number. d) √16 is a natural number, a whole number, an integer, and a rational number (it's 4). e) π (pi) is an irrational number.

Question 2: Is the number -2 a natural number, a whole number, or an integer? Explain your reasoning.

Answer 2: The number -2 is an integer. It is not a natural number (since natural numbers are positive integers), and it is not a whole number (since whole numbers include zero, and -2 is less than zero).

Question 3: Which of the following numbers is an irrational number?

a) 0.75 b) √25 c) 2/3 d) 0.333...

Answer 3: b) √25 is an irrational number. It cannot be expressed as a fraction of two integers, and it's not a repeating decimal.

Question 4: Identify the category (natural, whole, integer, rational, or irrational) that each of the following numbers belongs to:

a) -5,000 b) 3/4 c) √81 d) 0.125

Answer 4: a) -5,000 is an integer and a whole number. b) 3/4 is a rational number. c) √81 is a natural number, a whole number, an integer, and a rational number (it's 9). d) 0.125 is a rational number.

Question 5: True or False: All integers are rational numbers. Explain your answer.

Answer 5: True. All integers are indeed rational numbers because they can be expressed as the quotient of two integers (the numerator is the integer itself, and the denominator is 1). For example, -5 can be expressed as -5/1, which is a fraction of two integers.

In mathematics, various sets of numbers are organized hierarchically as subsets. This hierarchy helps us categorize and understand the relationships between different types of numbers. Here's an overview of how these sets of numbers are related as subsets:

1. Natural Numbers (N): Natural numbers are the set of positive counting numbers starting from 1 and going on infinitely (1, 2, 3, 4, ...). They do not include zero or negative numbers.

2. Whole Numbers (W): Whole numbers include all natural numbers and zero (0, 1, 2, 3, 4, ...). In other words, whole numbers are a superset of natural numbers.

3. Integers (Z): Integers include all positive and negative whole numbers along with zero (..., -3, -2, -1, 0, 1, 2, 3, ...). Integers are a superset of both natural and whole numbers.

4. Rational Numbers (Q): Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero. This set includes fractions, integers, and whole numbers as subsets.

5. Irrational Numbers: Irrational numbers are real numbers that cannot be expressed as fractions of two integers. They have non-repeating, non-terminating decimal expansions. Examples include the square root of 2 (√2), pi (π), and e (Euler's number). Irrational numbers are a complement to rational numbers and do not have a direct subset within the rational numbers.

6. Real Numbers (R): Real numbers are the complete set of all possible numbers, including both rational and irrational numbers. Real numbers encompass all the previously mentioned sets.

Real Numbers (R) | Rational Numbers (Q) |---------------------| | Irrational Numbers Whole Numbers (W) (√2, π, e, ...) | Natural Numbers (N)

In summary, real numbers include both rational and irrational numbers. Rational numbers include fractions, integers, and whole numbers, while integers encompass natural and whole numbers. Understanding these subsets is essential in various mathematical contexts and allows us to work with different types of numbers in a structured manner.

1. Natural Numbers (N):

   • Definition: Natural numbers are the set of positive counting numbers, starting from 1 and going on infinitely (1, 2, 3, 4, ...).
   • Example: N = {1, 2, 3, 4, ...}

2. Whole Numbers (W):

   • Definition: Whole numbers include all natural numbers along with zero (0, 1, 2, 3, 4, ...).
   • Example: W = {0, 1, 2, 3, 4, ...}

3. Integers (Z):

   • Definition: Integers include all positive and negative whole numbers along with zero (..., -3, -2, -1, 0, 1, 2, 3, ...).
   • Example: Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}

4. Rational Numbers (Q):

   • Definition: Rational numbers are numbers that can be expressed as the quotient or fraction of two integers, where the denominator is not zero.
   • Example: Q includes fractions (e.g., 1/2), integers (e.g., -3), and whole numbers (e.g., 5).

5. Irrational Numbers:

   • Definition: Irrational numbers are real numbers that cannot be expressed as fractions of two integers. They have non-repeating, non-terminating decimal expansions.
   • Example: √2, π (pi), and e (Euler's number) are examples of irrational numbers.

6. Real Numbers (R):

   • Definition: Real numbers are the complete set of all possible numbers, including both rational and irrational numbers.
   • Example: R encompasses all real numbers, such as 0.5, -3, √2, π, and more.

7. Complex Numbers (C):

• Definition: Complex numbers are numbers that have both a real part and an imaginary part. They are written in the form $�+��$, where $�$ and $�$ are real numbers, and $�$ is the imaginary unit (${�}^2=-1$).
• Example: $3+4�$ is a complex number.

8. Question 1: Classify each of the following numbers into the appropriate set: natural numbers (N), whole numbers (W), integers (Z), rational numbers (Q), or irrational numbers.

   a) -3 b) 0 c) 2/3 d) √25 e) -π

   Answer 1: a) -3 is an integer (Z). b) 0 is a whole number (W) and an integer (Z). c) 2/3 is a rational number (Q). d) √25 is a natural number (N), a whole number (W), an integer (Z), and a rational number (Q). e) -π is an irrational number.

   Question 2: Determine whether each of the following numbers is rational or irrational.

   a) 0.25 b) √4 c) -5/7 d) √3 e) 1.333...

   Answer 2: a) 0.25 is a rational number. b) √4 is a rational number. c) -5/7 is a rational number. d) √3 is an irrational number. e) 1.333... is a rational number.

   Question 3: Are the following statements true or false?

   a) All natural numbers are whole numbers. b) Zero is an irrational number. c) The square root of 49 is a rational number. d) All integers are rational numbers. e) Irrational numbers have repeating decimal expansions.

   Answer 3: a) True. All natural numbers are also whole numbers. b) False. Zero is not an irrational number; it's a whole number and an integer. c) True. The square root of 49 is a natural number and, therefore, a rational number. d) True. All integers are rational numbers because they can be expressed as fractions with denominators of 1. e) False. Irrational numbers have non-repeating decimal expansions.

   Question 4: Determine whether the following numbers are real or complex.

   a) -5 b) 2 + 3i c) √(-9) d) 1.61803398875...

   Answer 4: a) -5 is a real number. b) 2 + 3i is a complex number. c) √(-9) is a complex number (it has an imaginary part). d) 1.61803398875... is a real number.

9. The order of operations, often remembered by the acronym PEMDAS, stands for:

   1. P: Parentheses - Perform calculations inside parentheses first.
   2. E: Exponents - Evaluate expressions involving exponents (powers and roots).
   3. MD: Multiplication and Division - Perform multiplication and division from left to right.
   4. AS: Addition and Subtraction - Perform addition and subtraction from left to right.

   Let's perform some calculations using the order of operations:

   Example 1: Solve the expression: $3+4\times (2^2-1)$

   1. Start with parentheses: $2^2=4$.
   2. Subtract 1 inside the parentheses: $4-1=3$.
   3. Now, the expression becomes: $3+4\times 3$.
   4. Next, perform multiplication: $4\times 3=12$.
   5. Finally, perform addition: $3+12=15$.

   So, $3+4\times (2^2-1)=15$.

   Example 2: Solve the expression: $\frac{6+12}{3}\times 2$

   1. Start with parentheses: $6+12=18$.
   2. Now, the expression becomes: $\frac{18}{3}\times 2$.
   3. Perform division: $\frac{18}{3}=6$.
   4. Finally, perform multiplication: $6\times 2=12$.

   So, $\frac{6+12}{3}\times 2=12$.

   Example 3: Solve the expression: $5+2^3÷(6-4)$

   1. Start with parentheses: $6-4=2$.
   2. Next, evaluate the exponent: $2^3=8$.
   3. Now, the expression becomes: $5+8÷2$.
   4. Perform division: $8÷2=4$.
   5. Finally, perform addition: $5+4=9$.

   So, $5+2^3÷(6-4)=9$.

   Example 4: Evaluate $\sqrt{9}+\frac{12}{3}-2^2$.

   1. Start with the square root: $\sqrt{9}=3$.

   2. Then, perform the division: $\frac{12}{3}=4$.

   3. Calculate the exponent: $2^2=4$.

   4. Finally, subtract the results: $3+4-4=3$.

   So, $\sqrt{9}+\frac{12}{3}-2^2$ equals 3.

10. Example 5: Evaluate $2\cdot (4-1{)}^2+\frac{15}{3}$.

    1. Start inside the parentheses: $4-1=3$.

    2. Square the result: $3^2=9$.

    3. Multiply by 2: $2\cdot 9=18$.

    4. Divide 15 by 3: $\frac{15}{3}=5$.

    5. Add the results from steps 3 and 4: $18+5=23$.

    So, $2\cdot (4-1{)}^2+\frac{15}{3}$ equals 23.

    Example 6: Evaluate $\frac{2^3+5\cdot 4}{3-(7-4)}$.

    1. Calculate the exponent: $2^3=2\cdot 2\cdot 2=8$.

    2. Multiply 5 by 4: $5\cdot 4=20$.

    3. Calculate the expression inside the innermost parentheses: $7-4=3$.

    4. Subtract the result from step 3 from 3: $3-3=0$.

    5. Now, we can rewrite the expression as $\frac{8+20}{3-0}$.

    6. Perform the addition: $8+20=28$.

    7. Divide 3 by 0: Division by 0 is undefined in mathematics.

    So, $\frac{2^3+5\cdot 4}{3-(7-4)}$ is undefined due to division by 0.

11. These examples illustrate the importance of following the order of operations carefully to obtain the correct results, and they also show how to handle cases where division by 0 leads to undefined results.

12. Here's an example of simplifying a mathematical expression using the order of operations (PEMDAS):

    Expression: $2\cdot (3+5)-\frac{12}{4}$

    Let's simplify this step by step:

    1. Parentheses:

       • Inside the first set of parentheses, calculate $3+5=8$.

       So, the expression becomes $2\cdot 8-\frac{12}{4}$.

    2. Exponents (None in this case).

    3. Multiplication and Division (from left to right):

       • First, perform the multiplication: $2\cdot 8=16$.
       • Then, perform the division: $\frac{12}{4}=3$.

       So, the expression now becomes $16-3$.

    4. Addition and Subtraction (from left to right):

       • Finally, perform the subtraction: $16-3=13$.

    The simplified result of the expression $2\cdot (3+5)-\frac{12}{4}$ is $13$.

13. Here are three example questions, each followed by a step-by-step solution using the order of operations (PEMDAS):

    Example 1: Simplify the expression $4+3\cdot (6-2)$.

    Solution 1:

    1. Parentheses:

       • Inside the parentheses, calculate $6-2=4$.

       So, the expression becomes $4+3\cdot 4$.

    2. Exponents (None in this case).

    3. Multiplication and Division (from left to right):

       • First, perform the multiplication: $3\cdot 4=12$.

       So, the expression now becomes $4+12$.

    4. Addition and Subtraction (from left to right):

       • Finally, perform the addition: $4+12=16$.

    The simplified result of the expression $4+3\cdot (6-2)$ is $16$.

    Example 2: Simplify the expression $\frac{3+9}{2}-5\cdot 2$.

    Solution 2:

    1. Parentheses:

       • Inside the first set of parentheses, calculate $3+9=12$.

       So, the expression becomes $\frac{12}{2}-5\cdot 2$.

    2. Exponents (None in this case).

    3. Multiplication and Division (from left to right):

       • First, perform the division: $\frac{12}{2}=6$.
       • Then, perform the multiplication: $5\cdot 2=10$.

       So, the expression now becomes $6-10$.

    4. Addition and Subtraction (from left to right):

       • Finally, perform the subtraction: $6-10=-4$.

    The simplified result of the expression $\frac{3+9}{2}-5\cdot 2$ is $-4$.

    Example 3: Simplify the expression $5^2-(3\cdot 4-7)$.

    Solution 3:

    1. Parentheses:

       • Inside the parentheses, calculate $3\cdot 4-7=5$.

       So, the expression becomes $5^2-5$.

    2. Exponents:

       • Calculate $5^2=25$.

       So, the expression now becomes $25-5$.

    3. Multiplication and Division (from left to right):

       • Perform the subtraction: $25-5=20$.

    The simplified result of the expression $5^2-(3\cdot 4-7)$ is $20$.

    These examples demonstrate how to use the order of operations (PEMDAS) to simplify expressions step by step, ensuring that you obtain the correct results by following the prescribed sequence of operations.

14. Properties of real numbers are useful rules or characteristics that help us manipulate and simplify mathematical expressions and equations. Here are some common properties of real numbers, along with examples of how to use them:

    1. Commutative Property of Addition:

       • This property states that the order of adding real numbers doesn't affect the result.
       • $�+�=�+�$

       Example: $2+3=3+2=5$

    2. Commutative Property of Multiplication:

       • This property states that the order of multiplying real numbers doesn't affect the result.
       • $�\cdot �=�\cdot �$

       Example: $4\cdot 5=5\cdot 4=20$

    3. Associative Property of Addition:

       • This property states that the grouping of real numbers when adding doesn't affect the result.
       • $(�+�)+�=�+(�+�)$

       Example: $(2+3)+4=2+(3+4)=9$

    4. Associative Property of Multiplication:

       • This property states that the grouping of real numbers when multiplying doesn't affect the result.
       • ((a \⋅ b) \c⋅ c = a \c⋅ (b \c⋅ c)\

       Example: $(2\cdot 3)\cdot 4=2\cdot (3\cdot 4)=24$

    5. Distributive Property:

       • This property states that multiplication distributes over addition or subtraction.
       • $�\cdot (�+�)=�\cdot �+�\cdot �$

       Example: $2\cdot (3+4)=2\cdot 3+2\cdot 4=14$

    6. Inverse Property of Addition:

       • Every real number has an additive inverse, which, when added to the number, results in 0.
       • $�+(-�)=0$

       Example: $5+(-5)=0$

    7. Identity Property of Addition:

       • The identity element for addition is 0 because adding 0 to any real number doesn't change its value.
       • $�+0=�$

       Example: $7+0=7$

    8. Inverse Property of Multiplication:

       • Every nonzero real number has a multiplicative inverse (reciprocal), which, when multiplied, results in 1.
       • $�\cdot \frac{1}{�}=1$

       Example: $3\cdot \frac{1}{3}=1$

    9. Identity Property of Multiplication:

       • The identity element for multiplication is 1 because multiplying any real number by 1 doesn't change its value.
       • $�\cdot 1=�$

       Example: $6\cdot 1=6$

    By using these properties of real numbers, you can simplify expressions and equations, manipulate terms, and make calculations more efficient and manageable.

15. Here are some important properties of real numbers:

    1. Commutative Property of Addition:

       • Changing the order of adding two real numbers does not affect the result.
       • Example: $�+�=�+�$

    2. Commutative Property of Multiplication:

       • Changing the order of multiplying two real numbers does not affect the result.
       • Example: $�\cdot �=�\cdot �$

    3. Associative Property of Addition:

       • The grouping of real numbers when adding does not affect the result.
       • Example: $(�+�)+�=�+(�+�)$

    4. Associative Property of Multiplication:

       • The grouping of real numbers when multiplying does not affect the result.
       • Example: $(�\cdot �)\cdot �=�\cdot (�\cdot �)$

    5. Distributive Property:

       • Multiplication distributes over addition (or subtraction) of real numbers.
       • Example: $�\cdot (�+�)=�\cdot �+�\cdot �$

    6. Identity Property of Addition:

       • The identity element for addition is 0 because adding 0 to any real number does not change its value.
       • Example: $�+0=�$

    7. Identity Property of Multiplication:

       • The identity element for multiplication is 1 because multiplying any real number by 1 does not change its value.
       • Example: $�\cdot 1=�$

    8. Inverse Property of Addition:

       • Every real number has an additive inverse (negation), which, when added, results in 0.
       • Example: $�+(-�)=0$

    9. Inverse Property of Multiplication:

       • Every nonzero real number has a multiplicative inverse (reciprocal), which, when multiplied, results in 1.
       • Example: $�\cdot \frac{1}{�}=1$ (for $�\mathrm{\ne }0$)

    10. Addition of Zero:

        • Adding zero to any real number does not change its value.
        • Example: $�+0=�$

    11. Multiplication by Zero:

        • Multiplying any real number by zero results in zero.
        • Example: $�\cdot 0=0$ for any $�$

    12. Multiplication by One:

        • Multiplying any real number by one does not change its value.
        • Example: $�\cdot 1=�$

    These properties are fundamental in mathematics and are used to simplify expressions, manipulate equations, and solve various mathematical problems. They provide a set of rules that ensure consistency and coherence in mathematical operations involving real numbers.

16. Let's work through an example step by step, using properties of real numbers to simplify an expression:

    Example: Simplify the expression $3(4�-2)+5(2�-1)$.

    Solution:

    Step 1: Distribute (Apply the Distributive Property)

    Distribute the constants outside the parentheses to each term inside the parentheses.

    $3\cdot 4�-3\cdot 2+5\cdot 2�-5\cdot 1$

    Step 2: Simplify within the Parentheses

    Perform the multiplications within the parentheses.

    $12�-6+10�-5$

    Step 3: Combine Like Terms (Apply the Commutative and Associative Properties of Addition)

    Combine the like terms, which means combining terms with the same variable, $�$.

    $(12�+10�)+(-6-5)$

    Step 4: Perform Addition

    Perform the additions of coefficients.

    $22�-11$

    So, the simplified expression is $22�-11$.

    In this example, we used the distributive property to distribute the constants to terms within parentheses, then combines like terms, and applied the properties of addition to simplify the expression step by step. The simplified expression is $22�-11$.

17. These examples demonstrate how to substitute values into algebraic expressions and then simplify the expressions to obtain numerical results.

    Expression 1: Evaluate $3�+2�$ when $�=4$ and $�=5$.

    1. Replace $�$ with 4 and $�$ with 5 in the expression: $3�+2�=3(4)+2(5)$

    2. Perform the multiplications: $3(4)=12$ $2(5)=10$

    3. Add the results: $12+10=22$

    So, $3�+2�$ when $�=4$ and $�=5$ is equal to 22.

    Expression 2: Evaluate $2{�}^2-3�$ when $�=3$ and $�=2$.

    1. Replace $�$ with 3 and $�$ with 2 in the expression: $2{�}^2-3�=2(3^2)-3(2)$

    2. Perform the calculations inside the parentheses: $3^2=9$ $2(9)=18$ $3(2)=6$

    3. Subtract the results: $18-6=12$

    So, $2{�}^2-3�$ when $�=3$ and $�=2$ is equal to 12.

    Expression 3: Evaluate $\frac{5�}{2�}$ when $�=6$ and $�=3$.

    1. Replace $�$ with 6 and $�$ with 3 in the expression: $\frac{5�}{2�}=\frac{5(6)}{2(3)}$

    2. Perform the multiplications: $5(6)=30$ $2(3)=6$

    3. Divide the results: $\frac{30}{6}=5$

    So, $\frac{5�}{2�}$ when $�=6$ and $�=3$ is equal to 5.

    
    

    Example 4: Evaluate the expression $2�+3$ when $�=4$.

    Solution 4:

    1. Substitute the value of $�$ into the expression: $2�+3=2(4)+3$

    2. Perform the multiplications and additions: $2(4)+3=8+3=11$

    So, when $�=4$, the expression $2�+3$ equals 11.

    Example 5: Evaluate the expression $\frac{3�}{2}$ when $�=6$.

    Solution 5:

    1. Substitute the value of $�$ into the expression: $\frac{3�}{2}=\frac{3(6)}{2}$

    2. Perform the multiplications and division: $\frac{3(6)}{2}=\frac{18}{2}=9$

    So, when $�=6$, the expression $\frac{3�}{2}$ equals 9.

    Example 6: Evaluate the expression ${�}^2-4$ when $�=-2$.

    Solution 6:

    1. Substitute the value of $�$ into the expression: ${�}^2-4=(-2{)}^2-4$

    2. Perform the exponentiation and subtraction: $(-2{)}^2-4=4-4=0$

    So, when $�=-2$, the expression ${�}^2-4$ equals 0.

    These examples demonstrate how to evaluate algebraic expressions by substituting specific values for variables and performing the necessary arithmetic operations to obtain the result.

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19. Algebraic expressions are mathematical expressions that consist of variables, constants, and mathematical operations. They are used to represent relationships, describe patterns, and perform calculations in algebra and other branches of mathematics. Here are key components and characteristics of algebraic expressions:

    1. Variables: Variables are symbols that represent unknown or varying quantities. Common variable names include $�$, $�$, $�$, and $�$. Variables can take on different values, and algebraic expressions often involve manipulating these variables to solve equations or represent relationships.

    2. Constants: Constants are fixed numerical values that do not change. Examples of constants include numbers like 2, 3, -5, $�$, and $\sqrt{2}$. Constants can be used in algebraic expressions as coefficients or values to be operated on.

    3. Operators (Mathematical Operations): Algebraic expressions involve various mathematical operations, including addition, subtraction, multiplication, division, exponentiation, and root extraction. These operations are used to combine variables and constants to create complex expressions.

    4. Terms: Terms are the building blocks of algebraic expressions. They can be individual variables, constants, or combinations of both, separated by mathematical operators. For example, in the expression $3{�}^2-2�+7$, the terms are $3{�}^2$, $-2�$, and $7$.

    5. Coefficients: Coefficients are the numerical factors that multiply variables in a term. In the term $3{�}^2$, the coefficient is 3, and in $-2�$, the coefficient is -2.

    6. Exponents and Powers: Exponents are used to indicate repeated multiplication. Variables with exponents are often called powers. For example, in ${�}^2$, the exponent is 2, and the term represents "x squared."

    7. Variables with Coefficients: Expressions often involve variables with coefficients, such as $2�$ or $-3�$, where the coefficient multiplies the variable.

    8. Combining Like Terms: Algebraic expressions can be simplified by combining like terms, which are terms that have the same variable and exponent. For example, in $3{�}^2-2{�}^2$, the like terms $3{�}^2$ and $-2{�}^2$ can be combined to ${�}^2$.

    9. Polynomials: Polynomials are algebraic expressions that consist of one or more terms, where each term is a constant or a variable raised to a non-negative integer exponent. Examples include $2{�}^2-5�+3$ and ${�}^3+2{�}^2-�$.

    10. Equations: Algebraic expressions can be used to create equations, which are statements asserting that two expressions are equal. Solving equations involves finding values of the variables that make the equation true.

    Algebraic expressions are essential in mathematics and various scientific fields for modeling real-world phenomena, solving equations, and making predictions. They provide a powerful way to represent mathematical relationships and perform calculations.

20. Let's describe an algebraic expression and then provide an example with a step-by-step solution:

    Algebraic Expression: $2{�}^2-3�+4$

    Description: This algebraic expression consists of three terms. Each term has a coefficient and may have a variable raised to a power (exponent). Here's a breakdown:

    • Term 1: $2{�}^2$

      • Coefficient: 2
      • Variable: $�$
      • Exponent: 2
      • Description: This term is the product of 2 and the square of $�$. It represents a quadratic term.

    • Term 2: $-3�$

      • Coefficient: -3
      • Variable: $�$
      • Exponent: 1 (implied, as $�$ is the same as ${�}^1$)
      • Description: This term is the product of -3 and $�$. It represents a linear term.

    • Term 3: 4

      • Coefficient: 4
      • Description: This term is a constant, which means it doesn't depend on any variable. It's simply the number 4.

    Now, let's work with an example using this algebraic expression:

    Example: Evaluate the expression $2{�}^2-3�+4$ when $�=3$ and $�=-2$.

    Solution:

    1. Substitute the values of $�$ and $�$ into the expression: $2(3^2)-3(-2)+4$

    2. Perform the calculations:

       • $3^2=9$ (Square of 3).
       • $-3(-2)=6$ (Multiplication of -3 and -2).

    3. Continue simplifying: $2\cdot 9-6+4$

    4. Perform the multiplications and additions:

       • $2\cdot 9=18$.
       • $18-6=12$.
       • $12+4=16$.

    So, when $�=3$ and $�=-2$, the expression $2{�}^2-3�+4$ evaluates to 16.

21. Let's evaluate an algebraic expression at different values using an example with step-by-step answers:

    Algebraic Expression: $3{�}^2-2�+5$

    Description: This algebraic expression consists of three terms, each with a coefficient, a variable, and an exponent (if applicable). Here's a breakdown:

    • Term 1: $3{�}^2$

      • Coefficient: 3
      • Variable: $�$
      • Exponent: 2
      • Description: This term represents the product of 3 and the square of $�$, making it a quadratic term.

    • Term 2: $-2�$

      • Coefficient: -2
      • Variable: $�$
      • Exponent: 1 (implied)
      • Description: This term represents the product of -2 and $�$, making it a linear term.

    • Term 3: 5

      • Coefficient: 5
      • Description: This term is a constant, meaning it doesn't depend on any variable. It's simply the number 5.

    Now, let's evaluate this expression at different values:

    Example: Evaluate the expression $3{�}^2-2�+5$ at the following values:

    • When $�=2$ and $�=3$.
    • When $�=-1$ and $�=1$.

    Solution for $�=2$ and $�=3$:

    1. Substitute the values of $�$ and $�$ into the expression: $3(2^2)-2(3)+5$

    2. Perform the calculations:

       • $2^2=4$ (Square of 2).
       • $3\cdot 4=12$ (Multiplication of 3 and 4).
       • $2\cdot 3=6$ (Multiplication of 2 and 3).

    3. Continue simplifying: $3\cdot 4-2\cdot 3+5$

    4. Perform the multiplications and additions:

       • $3\cdot 4=12$.
       • $2\cdot 3=6$.
       • $12-6=6$.
       • $6+5=11$.

    So, when $�=2$ and $�=3$, the expression $3{�}^2-2�+5$ evaluates to 11.

    Solution for $�=-1$ and $�=1$:

    1. Substitute the values of $�$ and $�$ into the expression: $3(-1{)}^2-2(1)+5$

    2. Perform the calculations:

       • $(-1{)}^2=1$ (Square of -1).
       • $3\cdot 1=3$ (Multiplication of 3 and 1).
       • $2\cdot 1=2$ (Multiplication of 2 and 1).

    3. Continue simplifying: $3\cdot 1-2\cdot 1+5$

    4. Perform the multiplications and additions:

       • $3\cdot 1=3$.
       • $2\cdot 1=2$.
       • $3-2=1$.
       • $1+5=6$.

    So, when $�=-1$ and $�=1$, the expression $3{�}^2-2�+5$ evaluates to 6.

    In summary, we evaluated the algebraic expression $3{�}^2-2�+5$ at different values for $�$ and $�$ and obtained the results of 11 and 6 for the respective values.

22. Let's evaluate an algebraic expression step by step with an example:

    Algebraic Expression: $4�-2�+7$

    Description: This algebraic expression consists of three terms, each with a coefficient, a variable, and an implied exponent of 1. Here's a breakdown:

    • Term 1: $4�$

      • Coefficient: 4
      • Variable: $�$
      • Implied Exponent: 1
      • Description: This term represents the product of 4 and $�$, making it a linear term.

    • Term 2: $-2�$

      • Coefficient: -2
      • Variable: $�$
      • Implied Exponent: 1
      • Description: This term represents the product of -2 and $�$, making it a linear term.

    • Term 3: 7

      • Coefficient: 7
      • Description: This term is a constant, meaning it doesn't depend on any variable. It's simply the number 7.

    Now, let's evaluate this expression at a specific value:

    Example: Evaluate the expression $4�-2�+7$ when $�=3$ and $�=2$.

    Solution:

    1. Substitute the values of $�$ and $�$ into the expression: $4(3)-2(2)+7$

    2. Perform the calculations:

       • $4\cdot 3=12$ (Multiplication of 4 and 3).
       • $2\cdot 2=4$ (Multiplication of 2 and 2).

    3. Continue simplifying: $12-4+7$

    4. Perform the additions and subtractions:

       • $12-4=8$.
       • $8+7=15$.

    So, when $�=3$ and $�=2$, the expression $4�-2�+7$ evaluates to 15.

    In summary, we substituted the values of $�$ and $�$ into the algebraic expression and performed the necessary calculations to obtain the result of 15.

23. Algebra involves various formulas and equations that are used to solve problems, model real-world situations, and make mathematical calculations. Here are some common algebraic formulas and equations:

    1. Linear Equation: A linear equation in one variable, $�$, has the form $��+�=0$, where $�$ and $�$ are constants. It can be solved for $�$ to find a solution.

       Example: $2�+3=7$ can be solved for $�$, resulting in $�=2$.

    2. Quadratic Equation: A quadratic equation in one variable, $�$, has the form $�{�}^2+��+�=0$, where $�$, $�$, and $�$ are constants. The quadratic formula is often used to find solutions:

       $�=\frac{-�\pm \sqrt{{�}^2-4��}}{2�}$

       Example: ${�}^2-4�+4=0$ can be solved using the quadratic formula.

    3. Distance Formula: The distance between two points $({�}_1,{�}_1)$ and $({�}_2,{�}_2)$ in a coordinate plane is given by:

       $�=\sqrt{({�}_2-{�}_1{)}^2+({�}_2-{�}_1{)}^2}$

       This formula is used to calculate distances in geometry and physics.

    4. Slope-Intercept Form: The equation of a line in slope-intercept form is $�=��+�$, where $�$ is the slope and $�$ is the y-intercept. It describes the relationship between $�$ and $�$ for a straight line.

       Example: $�=2�+3$ represents a line with a slope of 2 and a y-intercept of 3.

    5. Quadratic Formula: As mentioned earlier, the quadratic formula is used to solve quadratic equations of the form $�{�}^2+��+�=0$.

    6. Pythagorean Theorem: In a right triangle, the Pythagorean theorem states that the square of the length of the hypotenuse ($�$) is equal to the sum of the squares of the lengths of the other two sides ($�$ and $�$).

       ${�}^2={�}^2+{�}^2$

       This theorem is used extensively in geometry.

    7. Exponent Laws: There are various laws that govern the manipulation of exponents, such as the product rule (${�}^{�}\cdot {�}^{�}={�}^{�+�}$), the power rule (${�}^{�}\cdot {�}^{�}=(�\cdot �{)}^{�}$), and the quotient rule ($\frac{{�}^{�}}{{�}^{�}}={�}^{�-�}$).

    8. Logarithmic Laws: Logarithmic laws, such as the product rule (${\log }_{�}(��)={\log }_{�}(�)+{\log }_{�}(�)$) and the power rule (${\log }_{�}({�}^{�})=�\cdot {\log }_{�}(�)$), are used to manipulate logarithmic expressions.

    9. Binomial Theorem: The binomial theorem is used to expand expressions of the form $(�+�{)}^{�}$ and provides a formula for the coefficients in the expansion.

       Example: $(�+�{)}^3={�}^3+3{�}^2�+3�{�}^2+{�}^3$

    These are just a few examples of the many formulas and equations used in algebra. Algebraic formulas are essential tools for solving a wide range of mathematical problems and for understanding relationships between variables in various contexts.

24. Let's use a formula with an example and provide the answer step by step.

    Formula: The formula for the area ($�$) of a rectangle is given by:

    $�=\text{length}\times \text{width}$

    Example: Calculate the area of a rectangle with a length of 8 meters and a width of 5 meters.

    Solution:

    1. Substitute the given values into the formula: $�=8\text{meters}\times 5\text{meters}$

    2. Perform the multiplication: $�=40\text{square meters}$

    So, the area of the rectangle with a length of 8 meters and a width of 5 meters is 40 square meters.

    In this example, we used the formula for the area of a rectangle, substituted the values of the length and width, and performed the necessary multiplication to calculate the area.

25. Let's simplify an algebraic expression with an example and provide the answer step by step.

    Algebraic Expression: $2�+3�-5�+7$

    Description: This algebraic expression consists of several terms involving the variable $�$. To simplify it, we'll combine like terms.

    Solution:

    1. Combine Like Terms:

       • Terms with $�$: $2�+3�-5�$ (These terms have the same variable $�$.)
       • Constant Term: $7$

    2. Simplify each group of like terms:

       • $2�+3�-5�$ simplifies to $(2+3-5)�$.
       • The constant term $7$ remains unchanged.

    3. Further simplify the expression:

       • $(2+3-5)�$ simplifies to $0�$, which is equivalent to $0$.

    4. Write the final simplified expression:

       • $0�+7$ simplifies to just $7$.

    So, the simplified form of the algebraic expression $2�+3�-5�+7$ is $7$.

    In this example, we combine like terms to simplify the expression and found that the simplified result is $7$.

26. Let's simplify a formula with an example and provide the answer step by step.

    Formula: The formula for the area ($�$) of a triangle with base ($�$) and height ($ℎ$) is given by:

    $�=\frac{1}{2}\times �\times ℎ$

    Example: Simplify the formula for the area of a triangle by expressing it without fractions.

    Solution: To simplify the formula by expressing it without fractions, we can multiply both sides of the equation by 2 to eliminate the fraction:

    $2�=�\times ℎ$

    Now, the formula for the area of a triangle without fractions is $2�=�\times ℎ$.

    In this example, we simplified the formula for the area of a triangle by eliminating the fraction $\frac{1}{2}$ and expressing it as $2�=�\times ℎ$

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28.