Simplify The Expression:$\[ 8a + 7a^2b - 7ab - 8ab - A \\]

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of the rules of algebra. In this article, we will simplify the given expression: $8a + 7a^2b - 7ab - 8ab - a$. We will use various techniques such as combining like terms, factoring, and canceling out common factors to simplify the expression.

Understanding the Expression

The given expression is a combination of several terms, each containing variables and constants. To simplify the expression, we need to understand the rules of algebra, particularly the rules for combining like terms and factoring.

Like Terms

Like terms are terms that have the same variable(s) raised to the same power. In the given expression, we can identify the following like terms:

• $8a$ and $-a$ are like terms because they both contain the variable $a$ raised to the power of 1.
• $7a^2b$ is a like term by itself because it contains the variable $a$ raised to the power of 2 and the variable $b$ raised to the power of 1.
• $-7ab$ and $-8ab$ are like terms because they both contain the variables $a$ and $b$ raised to the power of 1.

Combining Like Terms

Now that we have identified the like terms, we can combine them to simplify the expression. We will start by combining the like terms $8a$ and $-a$.

Combining $8a$ and $-a$

To combine $8a$ and $-a$, we need to add their coefficients. The coefficient of $8a$ is 8, and the coefficient of $-a$ is -1. Adding these coefficients, we get:

$$8a + (-a) = (8 + (-1))a = 7a$$

So, the expression $8a + -a$ simplifies to $7a$.

Factoring Out Common Factors

Now that we have combined the like terms, we can factor out common factors from the remaining terms. We will start by factoring out the common factor $-7ab$ from the terms $-7ab$ and $-8ab$.

Factoring Out $-7ab$

To factor out $-7ab$, we need to divide each term by $-7ab$. Dividing $-7ab$ by $-7ab$, we get:

$$\frac{-7ab}{-7ab} = 1$$

Dividing $-8ab$ by $-7ab$, we get:

$$\frac{-8ab}{-7ab} = \frac{8}{7}$$

So, the expression $-7ab - 8ab$ simplifies to $-7ab(1 + \frac{8}{7})$.

Simplifying the Expression

Now that we have factored out the common factor $-7ab$, we can simplify the expression further. We will start by simplifying the term $-7ab(1 + \frac{8}{7})$.

Simplifying $-7ab(1 + \frac{8}{7})$

To simplify $-7ab(1 + \frac{8}{7})$, we need to multiply the terms inside the parentheses. Multiplying 1 by $\frac{8}{7}$, we get:

$$1 \times \frac{8}{7} = \frac{8}{7}$$

So, the expression $-7ab(1 + \frac{8}{7})$ simplifies to $-7ab(\frac{8}{7} + 1)$.

Final Simplification

Now that we have simplified the expression $-7ab(1 + \frac{8}{7})$, we can simplify the entire expression. We will start by simplifying the term $-7ab(\frac{8}{7} + 1)$.

Simplifying $-7ab(\frac{8}{7} + 1)$

To simplify $-7ab(\frac{8}{7} + 1)$, we need to multiply the terms inside the parentheses. Multiplying $\frac{8}{7}$ by 1, we get:

$$\frac{8}{7} \times 1 = \frac{8}{7}$$

Multiplying 1 by 1, we get:

$$1 \times 1 = 1$$

So, the expression $-7ab(\frac{8}{7} + 1)$ simplifies to $-7ab(\frac{8}{7} + 1)$.

Final Answer

Now that we have simplified the expression, we can write the final answer.

The final answer is: $\boxed{7a - 7ab(\frac{8}{7} + 1)}$

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of the rules of algebra. In this article, we simplified the given expression: $8a + 7a^2b - 7ab - 8ab - a$. We used various techniques such as combining like terms, factoring, and canceling out common factors to simplify the expression. The final answer is $7a - 7ab(\frac{8}{7} + 1)$.

Frequently Asked Questions

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable(s) raised to the same power. Unlike terms are terms that have different variables or different powers of the same variable.

Q: How do I combine like terms?

A: To combine like terms, you need to add their coefficients. The coefficient of a term is the number in front of the variable(s).

Q: What is factoring out common factors?

A: Factoring out common factors is a technique used to simplify expressions by dividing each term by a common factor.

Q: How do I simplify expressions using factoring?

A: To simplify expressions using factoring, you need to identify the common factors and divide each term by that factor.

References

• Algebraic Expressions
• Like Terms
• Factoring

Further Reading

• Simplifying Algebraic Expressions
• Combining Like Terms
• Factoring Out Common Factors
  

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Q&A: Simplifying Algebraic Expressions

In the previous article, we simplified the expression: $8a + 7a^2b - 7ab - 8ab - a$. We used various techniques such as combining like terms, factoring, and canceling out common factors to simplify the expression. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable(s) raised to the same power. Unlike terms are terms that have different variables or different powers of the same variable.

Example

• $2x$ and $3x$ are like terms because they both contain the variable $x$ raised to the power of 1.
• $2x$ and $3y$ are unlike terms because they contain different variables.

Q: How do I combine like terms?

A: To combine like terms, you need to add their coefficients. The coefficient of a term is the number in front of the variable(s).

Example

• $2x + 3x = (2 + 3)x = 5x$
• $2x + 3y = (2 + 3)y = 5y$

Q: What is factoring out common factors?

A: Factoring out common factors is a technique used to simplify expressions by dividing each term by a common factor.

Example

• $6x + 12x = 6x(1 + 2) = 6x(3) = 18x$
• $6x + 12y = 6x(1 + 2) = 6x(3) = 18x$

Q: How do I simplify expressions using factoring?

A: To simplify expressions using factoring, you need to identify the common factors and divide each term by that factor.

Example

• $6x + 12x = 6x(1 + 2) = 6x(3) = 18x$
• $6x + 12y = 6x(1 + 2) = 6x(3) = 18x$

Q: What is the order of operations when simplifying algebraic expressions?

A: The order of operations when simplifying algebraic expressions is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Example

• $2x + 3(4x - 2) = 2x + 12x - 6 = 14x - 6$

Q: How do I simplify expressions with negative coefficients?

A: To simplify expressions with negative coefficients, you need to multiply each term by -1.

Example

• $-2x + 3x = (-2 + 3)x = x$
• $-2x - 3x = (-2 - 3)x = -5x$

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change. A constant is a value that does not change.

Example

• $x$ is a variable because it can represent any value.
• $5$ is a constant because it does not change.

Q: How do I simplify expressions with fractions?

A: To simplify expressions with fractions, you need to multiply each term by the reciprocal of the fraction.

Example

• $\frac{2x}{3} + \frac{4x}{3} = \frac{2x + 4x}{3} = \frac{6x}{3} = 2x$
• $\frac{2x}{3} - \frac{4x}{3} = \frac{2x - 4x}{3} = \frac{-2x}{3}$

Q: What is the difference between a linear expression and a quadratic expression?

A: A linear expression is an expression with one variable raised to the power of 1. A quadratic expression is an expression with one variable raised to the power of 2.

Example

• $2x + 3$ is a linear expression because it has one variable raised to the power of 1.
• $x^2 + 2x + 3$ is a quadratic expression because it has one variable raised to the power of 2.

Q: How do I simplify expressions with absolute values?

A: To simplify expressions with absolute values, you need to consider both the positive and negative cases.

Example

• $|x| = x$ if $x \geq 0$
• $|x| = -x$ if $x < 0$

Q: What is the difference between a rational expression and an irrational expression?

A: A rational expression is an expression that can be written as a fraction of two polynomials. An irrational expression is an expression that cannot be written as a fraction of two polynomials.

Example

• $\frac{x}{y}$ is a rational expression because it can be written as a fraction of two polynomials.
• $\sqrt{x}$ is an irrational expression because it cannot be written as a fraction of two polynomials.

Q: How do I simplify expressions with roots?

A: To simplify expressions with roots, you need to consider the properties of the root.

Example

• $\sqrt{x^2} = x$ if $x \geq 0$
• $\sqrt{x^2} = -x$ if $x < 0$

Q: What is the difference between a polynomial expression and a non-polynomial expression?

A: A polynomial expression is an expression that can be written as a sum of terms, each of which is a product of variables and constants. A non-polynomial expression is an expression that cannot be written as a sum of terms, each of which is a product of variables and constants.

Example

• $x^2 + 2x + 3$ is a polynomial expression because it can be written as a sum of terms, each of which is a product of variables and constants.
• $\sin(x)$ is a non-polynomial expression because it cannot be written as a sum of terms, each of which is a product of variables and constants.

Q: How do I simplify expressions with trigonometric functions?

A: To simplify expressions with trigonometric functions, you need to consider the properties of the function.

Example

• $\sin(x + y) = \sin(x)\cos(y) + \cos(x)\sin(y)$
• $\cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y)$

Q: What is the difference between a real number and an imaginary number?

A: A real number is a number that can be written as a fraction of two integers. An imaginary number is a number that cannot be written as a fraction of two integers.

Example

• $3$ is a real number because it can be written as a fraction of two integers.
• $i$ is an imaginary number because it cannot be written as a fraction of two integers.

Q: How do I simplify expressions with complex numbers?

A: To simplify expressions with complex numbers, you need to consider the properties of the number.

Example

• $a + bi$ is a complex number because it has a real part $a$ and an imaginary part $bi$.
• $a - bi$ is a complex number because it has a real part $a$ and an imaginary part $-bi$.

Q: What is the difference between a matrix and a vector?

A: A matrix is a rectangular array of numbers. A vector is a one-dimensional array of numbers.

Example

• $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ is a matrix because it is a rectangular array of numbers.
• $\begin{bmatrix} 1 \\ 2 \end{bmatrix}$ is a vector because it is a one-dimensional array of numbers.

Q: How do I simplify expressions with matrices?

A: To simplify expressions with matrices, you need to consider the properties of the matrix.

Example

• $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$
• $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$

Q: What is the difference between a determinant and a matrix?

A: A determinant is a scalar value that can be calculated from a matrix. A matrix is a rectangular array of numbers.

Example

• $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ is a matrix because it is a rectangular array of numbers.
• $\det \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} = 1 \times 4 -