Rewrite Without Parentheses: ( 4 A 6 B 5 − 7 A 4 ) ( − 9 A B 3 (4a^6b^5 - 7a^4)(-9ab^3 ( 4 A 6 B 5 − 7 A 4 ) ( − 9 A B 3 ]Simplify Your Answer As Much As Possible.

Mar 8, 2025 by ADMIN 164 views

Simplifying Algebraic Expressions: A Step-by-Step Guide

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will focus on simplifying a specific type of algebraic expression, namely the product of two binomials. We will use the given expression $(4a^6b^5 - 7a^4)(-9ab^3)$ as an example and walk through the steps to simplify it.

Understanding the Expression

Before we start simplifying the expression, let's break it down and understand its components. The given expression is a product of two binomials:

$(4a^6b^5 - 7a^4)(-9ab^3)$

This expression consists of two parts: the first part is a binomial with two terms, and the second part is a binomial with two terms as well. To simplify this expression, we need to multiply each term in the first binomial by each term in the second binomial.

Distributive Property

To simplify the expression, we will use the distributive property, which states that for any real numbers $a$ , $b$ , and $c$ , the following equation holds:

$a(b + c) = ab + ac$

Using this property, we can rewrite the given expression as:

$4a^6b^5(-9ab^3) - 7a^4(-9ab^3)$

Multiplying Terms

Now, let's multiply each term in the first binomial by each term in the second binomial:

$4a^6b^5(-9ab^3) = -36a^7b^8$

$7a^4(-9ab^3) = -63a^5b^4$

Combining Like Terms

Now that we have multiplied each term, we can combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two terms with the variable $a$ raised to the power of 7 and two terms with the variable $a$ raised to the power of 5.

$-36a^7b^8 - 63a^5b^4$

Final Simplification

The final step is to simplify the expression by combining like terms. In this case, we have two terms with the variable $a$ raised to the power of 7 and two terms with the variable $a$ raised to the power of 5. However, there are no like terms with the variable $b$ , so we cannot combine them.

The final simplified expression is:

$-36a^7b^8 - 63a^5b^4$

Conclusion

In this article, we have walked through the steps to simplify a specific type of algebraic expression, namely the product of two binomials. We have used the distributive property to multiply each term in the first binomial by each term in the second binomial and then combined like terms to simplify the expression. The final simplified expression is $-36a^7b^8 - 63a^5b^4$ .

Tips and Tricks

When simplifying algebraic expressions, it's essential to use the distributive property to multiply each term in the first binomial by each term in the second binomial.
When combining like terms, make sure to identify the terms with the same variable raised to the same power.
When simplifying expressions, it's essential to be careful with the signs of the terms.

Common Mistakes

Failing to use the distributive property when multiplying terms.
Failing to combine like terms when simplifying expressions.
Making errors when multiplying terms, such as forgetting to multiply the coefficients or the variables.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications, including:

Calculating the area and perimeter of shapes.
Determining the volume of solids.
Modeling population growth and decline.
Solving systems of equations.

Final Thoughts

Simplifying algebraic expressions is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can simplify even the most complex expressions. Remember to use the distributive property, combine like terms, and be careful with the signs of the terms. With practice and patience, you will become proficient in simplifying algebraic expressions and be able to tackle even the most challenging problems.

Introduction

In our previous article, we walked through the steps to simplify a specific type of algebraic expression, namely the product of two binomials. We received many questions from readers who were struggling to understand the concepts and apply them to their own problems. In this article, we will address some of the most common questions and provide additional guidance to help you master the art of simplifying algebraic expressions.

Q&A

Q: What is the distributive property, and how do I use it to simplify expressions?

A: The distributive property is a fundamental concept in algebra that allows you to multiply each term in the first binomial by each term in the second binomial. To use it, simply multiply each term in the first binomial by each term in the second binomial, and then combine like terms.

Q: How do I identify like terms?

A: Like terms are terms that have the same variable raised to the same power. To identify like terms, look for terms that have the same variable and exponent. For example, $2x^2$ and $3x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

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

A: A coefficient is a number that is multiplied by a variable, while a variable is a letter or symbol that represents a value. For example, in the expression $2x$ , the 2 is the coefficient, and the $x$ is the variable.

Q: How do I simplify expressions with negative coefficients?

A: When simplifying expressions with negative coefficients, remember to multiply the negative coefficient by each term in the second binomial. For example, in the expression $-3x(2y)$ , the negative coefficient $-3$ is multiplied by each term in the second binomial, resulting in $-6xy$ .

Q: Can I simplify expressions with variables raised to different powers?

A: Yes, you can simplify expressions with variables raised to different powers. To do this, simply multiply each term in the first binomial by each term in the second binomial, and then combine like terms. For example, in the expression $(x^2 + 3x)(2x^3)$ , the variable $x$ is raised to different powers in each term. To simplify this expression, multiply each term in the first binomial by each term in the second binomial, resulting in $2x^5 + 6x^4$ .

Q: How do I simplify expressions with fractions?

A: When simplifying expressions with fractions, remember to multiply each term in the first binomial by each term in the second binomial, and then simplify the resulting fraction. For example, in the expression $(\frac{1}{2}x + \frac{1}{3}y)(\frac{2}{3}x)$ , the fractions are multiplied by each term in the second binomial, resulting in $\frac{1}{3}x^2 + \frac{1}{6}xy$ .

Tips and Tricks

When simplifying expressions, always use the distributive property to multiply each term in the first binomial by each term in the second binomial.
When combining like terms, make sure to identify the terms with the same variable raised to the same power.
When simplifying expressions with negative coefficients, remember to multiply the negative coefficient by each term in the second binomial.
When simplifying expressions with variables raised to different powers, simply multiply each term in the first binomial by each term in the second binomial, and then combine like terms.
When simplifying expressions with fractions, remember to multiply each term in the first binomial by each term in the second binomial, and then simplify the resulting fraction.

Common Mistakes

Failing to use the distributive property when multiplying terms.
Failing to combine like terms when simplifying expressions.
Making errors when multiplying terms, such as forgetting to multiply the coefficients or the variables.
Failing to simplify fractions when simplifying expressions.

Real-World Applications

Simplifying algebraic expressions has numerous real-world applications, including:

Calculating the area and perimeter of shapes.
Determining the volume of solids.
Modeling population growth and decline.
Solving systems of equations.