Simplify The Expression: $\left(6x - 7 + 2x^3\right) + \left(9x^3 + 4 + 5x^5\right$\]

Introduction

In algebra, simplifying expressions is a crucial skill that helps us to manipulate and solve equations. It involves combining like terms, removing unnecessary parentheses, and rearranging the terms to make the expression more manageable. In this article, we will focus on simplifying the given expression: $\left(6x - 7 + 2x^3\right) + \left(9x^3 + 4 + 5x^5\right)$. We will break down the steps involved in simplifying this expression and provide a clear explanation of each step.

Understanding the Expression

The given expression consists of two sets of parentheses, each containing a combination of variables and constants. The expression can be written as:

$\left(6x - 7 + 2x^3\right) + \left(9x^3 + 4 + 5x^5\right)$

To simplify this expression, we need to combine like terms, which means combining the terms that have the same variable and exponent.

Step 1: Identify Like Terms

The first step in simplifying the expression is to identify the like terms. In this expression, the like terms are:

• $6x$ and $9x^3$ (both have the variable $x$)
• $-7$ and $4$ (both are constants)
• $2x^3$ and $9x^3$ (both have the variable $x^3$)
• $5x^5$ (has the variable $x^5$)

Step 2: Combine Like Terms

Now that we have identified the like terms, we can combine them. To combine like terms, we add or subtract the coefficients of the terms.

• $6x + 9x = 15x$ (combining the $x$ terms)
• $-7 + 4 = -3$ (combining the constants)
• $2x^3 + 9x^3 = 11x^3$ (combining the $x^3$ terms)
• $5x^5$ remains the same (since there is no other term with the variable $x^5$)

Step 3: Simplify the Expression

Now that we have combined the like terms, we can simplify the expression by removing unnecessary parentheses and rearranging the terms.

$\left(6x - 7 + 2x^3\right) + \left(9x^3 + 4 + 5x^5\right)$

$= 15x - 3 + 11x^3 + 5x^5$

$= 5x^5 + 11x^3 + 15x - 3$

Conclusion

Simplifying the given expression involves combining like terms, removing unnecessary parentheses, and rearranging the terms. By following the steps outlined in this article, we can simplify the expression and arrive at the final answer. The simplified expression is:

$5x^5 + 11x^3 + 15x - 3$

This expression is now in a more manageable form, and we can use it to solve equations or manipulate algebraic expressions.

Tips and Tricks

• When simplifying expressions, always start by identifying like terms.
• Use the distributive property to remove unnecessary parentheses.
• Combine like terms by adding or subtracting the coefficients of the terms.
• Rearrange the terms to make the expression more manageable.

Common Mistakes to Avoid

• Failing to identify like terms can lead to incorrect simplification.
• Not using the distributive property can result in unnecessary parentheses.
• Not combining like terms correctly can lead to incorrect coefficients.

Real-World Applications

Simplifying expressions is a crucial skill in algebra, and it has many real-world applications. For example:

• In physics, simplifying expressions is used to solve equations of motion and energy.
• In engineering, simplifying expressions is used to design and optimize systems.
• In economics, simplifying expressions is used to model and analyze economic systems.

Conclusion

Simplifying the expression $\left(6x - 7 + 2x^3\right) + \left(9x^3 + 4 + 5x^5\right)$ involves combining like terms, removing unnecessary parentheses, and rearranging the terms. By following the steps outlined in this article, we can simplify the expression and arrive at the final answer. The simplified expression is:

$5x^5 + 11x^3 + 15x - 3$

Q&A: Simplifying Expressions

In this article, we will provide a Q&A section to help you better understand the concept of simplifying expressions. We will cover common questions and provide detailed answers to help you master this crucial skill in algebra.

Q: What is simplifying an expression?

A: Simplifying an expression involves combining like terms, removing unnecessary parentheses, and rearranging the terms to make the expression more manageable.

Q: Why is simplifying expressions important?

A: Simplifying expressions is important because it helps us to manipulate and solve equations. By simplifying expressions, we can make them easier to work with and solve.

Q: How do I identify like terms?

A: To identify like terms, look for terms that have the same variable and exponent. For example, $2x$ and $5x$ are like terms because they both have the variable $x$.

Q: How do I combine like terms?

A: To combine like terms, add or subtract the coefficients of the terms. For example, $2x + 5x = 7x$.

Q: What is the distributive property?

A: The distributive property is a rule that allows us to remove unnecessary parentheses by multiplying each term inside the parentheses by the coefficient outside the parentheses.

Q: How do I use the distributive property?

A: To use the distributive property, multiply each term inside the parentheses by the coefficient outside the parentheses. For example, $2(3x + 4) = 6x + 8$.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

• Failing to identify like terms
• Not using the distributive property
• Not combining like terms correctly
• Not rearranging the terms to make the expression more manageable

Q: How do I know when an expression is simplified?

A: An expression is simplified when there are no like terms left to combine and the terms are arranged in a logical and consistent order.

Q: Can you provide an example of a simplified expression?

A: Yes, here is an example of a simplified expression:

$2x^2 + 3x - 4$

This expression is simplified because there are no like terms left to combine and the terms are arranged in a logical and consistent order.

Q: How do I apply simplifying expressions in real-world scenarios?

A: Simplifying expressions is used in many real-world scenarios, including:

• Physics: Simplifying expressions is used to solve equations of motion and energy.
• Engineering: Simplifying expressions is used to design and optimize systems.
• Economics: Simplifying expressions is used to model and analyze economic systems.

Conclusion

Simplifying expressions is a crucial skill in algebra that has many real-world applications. By following the steps outlined in this article and practicing with examples, you can master this skill and apply it to a variety of scenarios.

Tips and Tricks

• Always start by identifying like terms.
• Use the distributive property to remove unnecessary parentheses.
• Combine like terms by adding or subtracting the coefficients of the terms.
• Rearrange the terms to make the expression more manageable.

Common Mistakes to Avoid

• Failing to identify like terms
• Not using the distributive property
• Not combining like terms correctly
• Not rearranging the terms to make the expression more manageable

Real-World Applications

Simplifying expressions is used in many real-world scenarios, including:

• Physics: Simplifying expressions is used to solve equations of motion and energy.
• Engineering: Simplifying expressions is used to design and optimize systems.
• Economics: Simplifying expressions is used to model and analyze economic systems.

Conclusion

Simplifying expressions is a crucial skill in algebra that has many real-world applications. By following the steps outlined in this article and practicing with examples, you can master this skill and apply it to a variety of scenarios.