Simplify The Expression:$ (7x 4y 5 + 3x 5y 4 + 7xy) + (5x 5y 4 - 5xy - 7x 4y 5) $

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will delve into the world of algebraic manipulation and explore the process of simplifying a given expression. We will use the expression $(7x^4y^5 + 3x^5y^4 + 7xy) + (5x^5y^4 - 5xy - 7x^4y^5)$ as a case study and demonstrate the step-by-step process of simplifying it.

Understanding the Expression

Before we begin simplifying the expression, let's take a closer look at it. The expression consists of two sets of parentheses, each containing a combination of terms with variables and coefficients. The variables in this expression are $x$ and $y$, and the coefficients are the numbers that multiply the variables.

(7x^4y^5 + 3x^5y^4 + 7xy) + (5x^5y^4 - 5xy - 7x^4y^5)

Distributive Property

To simplify the expression, we will use the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$. This property allows us to distribute the addition operation to each term inside the parentheses.

(7x^4y^5 + 3x^5y^4 + 7xy) + (5x^5y^4 - 5xy - 7x^4y^5)
= (7x^4y^5 + 5x^5y^4) + (3x^5y^4 - 5xy) + (7xy - 7x^4y^5)

Combining Like Terms

Now that we have distributed the addition operation, we can combine like terms. Like terms are terms that have the same variable(s) raised to the same power(s). In this expression, we have several like terms that we can combine.

(7x^4y^5 + 5x^5y^4) + (3x^5y^4 - 5xy) + (7xy - 7x^4y^5)
= (7x^4y^5 - 7x^4y^5) + (5x^5y^4 + 3x^5y^4) + (7xy - 5xy)

Simplifying the Expression

Now that we have combined like terms, we can simplify the expression further. We can eliminate the terms that cancel each other out and combine the remaining terms.

(7x^4y^5 - 7x^4y^5) + (5x^5y^4 + 3x^5y^4) + (7xy - 5xy)
= 0 + 8x^5y^4 + 2xy
= 8x^5y^4 + 2xy

Conclusion

In this article, we have demonstrated the process of simplifying a given algebraic expression using the distributive property and combining like terms. We have used the expression $(7x^4y^5 + 3x^5y^4 + 7xy) + (5x^5y^4 - 5xy - 7x^4y^5)$ as a case study and shown how to simplify it step by step. By following these steps, you can simplify any algebraic expression and gain a deeper understanding of the underlying mathematical concepts.

Frequently Asked Questions

• What is the distributive property? The distributive property is a mathematical property that allows us to distribute the addition operation to each term inside the parentheses.
• What are like terms? Like terms are terms that have the same variable(s) raised to the same power(s).
• How do I simplify an algebraic expression? To simplify an algebraic expression, you can use the distributive property and combine like terms.

Further Reading

• Algebraic Manipulation: A Comprehensive Guide
• Simplifying Algebraic Expressions: A Step-by-Step Guide
• Distributive Property: A Mathematical Property

References

• [1] Algebraic Manipulation: A Comprehensive Guide
• [2] Simplifying Algebraic Expressions: A Step-by-Step Guide
• [3] Distributive Property: A Mathematical Property
  

Introduction

Simplifying algebraic expressions is a crucial skill for any math enthusiast. In our previous article, we demonstrated the process of simplifying a given expression using the distributive property and combining like terms. However, we understand that sometimes, you may have questions or need further clarification on certain concepts. In this article, we will address some of the most frequently asked questions related to algebraic expression simplification.

Q&A

Q: What is the distributive property?

A: The distributive property is a mathematical property that allows us to distribute the addition operation to each term inside the parentheses. It states that for any real numbers $a$, $b$, and $c$, $a(b + c) = ab + ac$.

Q: What are like terms?

A: Like terms are terms that have the same variable(s) raised to the same power(s). For example, $2x^2$ and $5x^2$ are like terms because they both have the variable $x$ raised to the power of 2.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you can use the distributive property and combine like terms. Here's a step-by-step guide:

1. Distribute the addition operation to each term inside the parentheses.
2. Combine like terms by adding or subtracting the coefficients of the like terms.
3. Simplify the resulting expression by eliminating any terms that cancel each other out.

Q: What is the difference between combining like terms and simplifying an expression?

A: Combining like terms involves adding or subtracting the coefficients of the like terms, while simplifying an expression involves eliminating any terms that cancel each other out.

Q: Can I simplify an expression by canceling out terms?

A: Yes, you can simplify an expression by canceling out terms. However, you must make sure that the terms you are canceling out are like terms.

Q: What is the order of operations when simplifying an expression?

A: The order of operations when simplifying an expression is:

1. Evaluate any expressions inside parentheses.
2. Evaluate any exponential expressions.
3. Evaluate any multiplication and division operations from left to right.
4. Evaluate any addition and subtraction operations from left to right.

Q: Can I simplify an expression with variables in the denominator?

A: Yes, you can simplify an expression with variables in the denominator. However, you must make sure that the variables are not zero.

Q: What is the difference between simplifying an expression and solving an equation?

A: Simplifying an expression involves reducing the expression to its simplest form, while solving an equation involves finding the value of the variable that makes the equation true.

Conclusion

In this article, we have addressed some of the most frequently asked questions related to algebraic expression simplification. We hope that this Q&A guide has provided you with a better understanding of the concepts and has helped you to simplify algebraic expressions with confidence.

Frequently Asked Questions

• What is the distributive property?
• What are like terms?
• How do I simplify an algebraic expression?
• What is the difference between combining like terms and simplifying an expression?
• Can I simplify an expression by canceling out terms?
• What is the order of operations when simplifying an expression?
• Can I simplify an expression with variables in the denominator?
• What is the difference between simplifying an expression and solving an equation?

Further Reading

• Algebraic Manipulation: A Comprehensive Guide
• Simplifying Algebraic Expressions: A Step-by-Step Guide
• Distributive Property: A Mathematical Property

References

• [1] Algebraic Manipulation: A Comprehensive Guide
• [2] Simplifying Algebraic Expressions: A Step-by-Step Guide
• [3] Distributive Property: A Mathematical Property