Simplify The Expression:${ 3r 4(5r 2 - R + 4) + 12r^6 + 12 }$

Introduction to Simplifying Algebraic Expressions

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves combining like terms and removing any unnecessary components from the expression. In this article, we will focus on simplifying the given expression: $3r^4(5r^2 - r + 4) + 12r^6 + 12$. We will use various techniques to simplify the expression and provide a clear understanding of the process.

Understanding the Expression

The given expression is a combination of three terms: $3r^4(5r^2 - r + 4)$, $12r^6$, and $12$. To simplify the expression, we need to understand the properties of exponents and how to combine like terms.

Properties of Exponents

Exponents are a shorthand way of writing repeated multiplication. For example, $r^4$ means $r$ multiplied by itself four times: $r \times r \times r \times r$. When we multiply two or more terms with the same base, we add their exponents. For example, $r^4 \times r^2 = r^{4+2} = r^6$.

Combining Like Terms

Like terms are terms that have the same variable and exponent. For example, $3r^4$ and $12r^4$ are like terms because they both have the variable $r$ and the exponent $4$. When we combine like terms, we add their coefficients. For example, $3r^4 + 12r^4 = 15r^4$.

Simplifying the Expression

Now that we have a good understanding of the properties of exponents and how to combine like terms, we can simplify the given expression.

Distributing the Coefficient

The first step in simplifying the expression is to distribute the coefficient $3r^4$ to the terms inside the parentheses: $3r^4(5r^2 - r + 4)$. This means multiplying $3r^4$ by each term inside the parentheses: $3r^4 \times 5r^2$, $3r^4 \times -r$, and $3r^4 \times 4$.

3r^4(5r^2 - r + 4) = 3r^4 \times 5r^2 - 3r^4 \times r + 3r^4 \times 4

Simplifying the Terms

Now that we have distributed the coefficient, we can simplify each term.

3r^4 \times 5r^2 = 15r^6
3r^4 \times -r = -3r^5
3r^4 \times 4 = 12r^4

Combining Like Terms

Now that we have simplified each term, we can combine like terms. The terms $15r^6$ and $12r^6$ are like terms because they both have the variable $r$ and the exponent $6$. When we combine like terms, we add their coefficients: $15r^6 + 12r^6 = 27r^6$.

3r^4(5r^2 - r + 4) = 15r^6 - 3r^5 + 12r^4

Adding the Remaining Terms

Finally, we can add the remaining terms: $15r^6 - 3r^5 + 12r^4 + 12r^6 + 12$.

15r^6 - 3r^5 + 12r^4 + 12r^6 + 12 = 27r^6 - 3r^5 + 12r^4 + 12

Conclusion

In this article, we simplified the expression $3r^4(5r^2 - r + 4) + 12r^6 + 12$ using various techniques. We distributed the coefficient, simplified the terms, and combined like terms. The final simplified expression is $27r^6 - 3r^5 + 12r^4 + 12$. This expression is a combination of four terms: $27r^6$, $-3r^5$, $12r^4$, and $12$. We hope that this article has provided a clear understanding of the process of simplifying algebraic expressions.

Frequently Asked Questions

• Q: What is the final simplified expression? A: The final simplified expression is $27r^6 - 3r^5 + 12r^4 + 12$.
• Q: How do I simplify an algebraic expression? A: To simplify an algebraic expression, you need to distribute the coefficient, simplify the terms, and combine like terms.
• Q: What are like terms? A: Like terms are terms that have the same variable and exponent.

Further Reading

• Algebraic Expressions: A Comprehensive Guide
• Simplifying Algebraic Expressions: A Step-by-Step Guide
• Combining Like Terms: A Tutorial
  

Introduction to Simplifying Algebraic Expressions

Simplifying algebraic expressions is a crucial skill in mathematics, particularly in algebra and calculus. It involves combining like terms and removing any unnecessary components from the expression. In this article, we will focus on simplifying the given expression: $3r^4(5r^2 - r + 4) + 12r^6 + 12$. We will use various techniques to simplify the expression and provide a clear understanding of the process.

Q&A: Simplifying Algebraic Expressions

Q: What is the final simplified expression?

A: The final simplified expression is $27r^6 - 3r^5 + 12r^4 + 12$.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to follow these steps:

1. Distribute the coefficient to the terms inside the parentheses.
2. Simplify the terms by multiplying the coefficient with each term.
3. Combine like terms by adding their coefficients.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, $3r^4$ and $12r^4$ are like terms because they both have the variable $r$ and the exponent $4$.

Q: How do I combine like terms?

A: To combine like terms, you need to add their coefficients. For example, $3r^4 + 12r^4 = 15r^4$.

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

A: A coefficient is a number that is multiplied with a variable. For example, in the term $3r^4$, the coefficient is $3$ and the variable is $r^4$. A variable is a letter or symbol that represents a value.

Q: How do I simplify an expression with multiple variables?

A: To simplify an expression with multiple variables, you need to follow the same steps as before: distribute the coefficient, simplify the terms, and combine like terms.

Q: What is the importance of simplifying algebraic expressions?

A: Simplifying algebraic expressions is important because it helps to:

• Reduce the complexity of the expression
• Make it easier to solve equations and inequalities
• Improve the accuracy of calculations

Real-World Applications of Simplifying Algebraic Expressions

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

• Physics: Simplifying algebraic expressions is used to describe the motion of objects and the behavior of physical systems.
• Engineering: Simplifying algebraic expressions is used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Simplifying algebraic expressions is used to model economic systems and make predictions about future trends.

Conclusion

In this article, we simplified the expression $3r^4(5r^2 - r + 4) + 12r^6 + 12$ using various techniques. We also provided a Q&A section to answer common questions about simplifying algebraic expressions. We hope that this article has provided a clear understanding of the process of simplifying algebraic expressions and its importance in real-world applications.

Frequently Asked Questions

• Q: What is the final simplified expression? A: The final simplified expression is $27r^6 - 3r^5 + 12r^4 + 12$.
• Q: How do I simplify an algebraic expression? A: To simplify an algebraic expression, you need to distribute the coefficient, simplify the terms, and combine like terms.
• Q: What are like terms? A: Like terms are terms that have the same variable and exponent.

Further Reading

• Algebraic Expressions: A Comprehensive Guide
• Simplifying Algebraic Expressions: A Step-by-Step Guide
• Combining Like Terms: A Tutorial