Expand And Simplify 4 ( 7 5 − 3 4(7 \sqrt{5} - 3 4 ( 7 5 ​ − 3 ].

Introduction

Algebraic expressions are a fundamental concept in mathematics, and expanding and simplifying them is a crucial skill to master. In this article, we will focus on expanding and simplifying the given expression $4(7 \sqrt{5} - 3)$. We will break down the process into manageable steps, making it easier to understand and apply.

Understanding the Expression

Before we begin, let's take a closer look at the given expression. The expression is $4(7 \sqrt{5} - 3)$, where $4$ is a coefficient, $7 \sqrt{5}$ is a term with a square root, and $3$ is a constant term. The expression is enclosed in parentheses, indicating that it is a single unit.

Step 1: Distributing the Coefficient

To expand the expression, we need to distribute the coefficient $4$ to each term inside the parentheses. This means multiplying $4$ by each term separately.

Distributing the Coefficient to the First Term

To distribute the coefficient to the first term, we multiply $4$ by $7 \sqrt{5}$.

4 \cdot 7 \sqrt{5} = 28 \sqrt{5}

Distributing the Coefficient to the Second Term

To distribute the coefficient to the second term, we multiply $4$ by $-3$.

4 \cdot -3 = -12

Step 2: Combining Like Terms

Now that we have distributed the coefficient to each term, we can combine like terms. In this case, we have two terms: $28 \sqrt{5}$ and $-12$. Since they are like terms, we can combine them by adding or subtracting their coefficients.

28 \sqrt{5} - 12

Step 3: Simplifying the Expression

The expression is now simplified, and we can write it in its final form.

28 \sqrt{5} - 12

Conclusion

Expanding and simplifying algebraic expressions is an essential skill in mathematics. By following the steps outlined in this article, we can easily expand and simplify expressions like $4(7 \sqrt{5} - 3)$. Remember to distribute the coefficient to each term, combine like terms, and simplify the expression to get the final result.

Common Mistakes to Avoid

When expanding and simplifying algebraic expressions, there are several common mistakes to avoid.

• Not distributing the coefficient correctly: Make sure to distribute the coefficient to each term separately.
• Not combining like terms correctly: Combine like terms by adding or subtracting their coefficients.
• Not simplifying the expression correctly: Simplify the expression by combining like terms and removing any unnecessary parentheses.

Practice Problems

To practice expanding and simplifying algebraic expressions, try the following problems:

• Expand and simplify the expression $3(2x + 5)$.
• Expand and simplify the expression $2(3x - 2)$.
• Expand and simplify the expression $4(2x + 3)$.

Real-World Applications

Expanding and simplifying algebraic expressions has numerous real-world applications. Some examples include:

• Science and Engineering: Algebraic expressions are used to model real-world phenomena, such as the motion of objects and the behavior of electrical circuits.
• Economics: Algebraic expressions are used to model economic systems, such as supply and demand curves.
• Computer Science: Algebraic expressions are used to model algorithms and data structures.

Final Thoughts

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Introduction

In our previous article, we discussed the steps involved in expanding and simplifying algebraic expressions. In this article, we will answer some frequently asked questions (FAQs) related to expanding and simplifying algebraic expressions.

Q: What is the difference between expanding and simplifying an algebraic expression?

A: Expanding an algebraic expression involves distributing a coefficient to each term inside the parentheses, while simplifying an algebraic expression involves combining like terms and removing any unnecessary parentheses.

Q: How do I distribute a coefficient to each term inside the parentheses?

A: To distribute a coefficient to each term inside the parentheses, multiply the coefficient by each term separately. For example, if we have the expression $4(7 \sqrt{5} - 3)$, we would multiply $4$ by $7 \sqrt{5}$ and $4$ by $-3$.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, $2x$ and $5x$ are like terms because they both have the variable $x$ raised to the power of $1$.

Q: How do I combine like terms?

A: To combine like terms, add or subtract their coefficients. For example, if we have the expression $2x + 5x$, we would combine the like terms by adding their coefficients: $2x + 5x = 7x$.

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

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

1. Evaluate any expressions inside parentheses.
2. Exponentiate any terms with exponents.
3. Multiply and divide any terms from left to right.
4. Add and subtract any terms from left to right.

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

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

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

A: Some common mistakes to avoid when expanding and simplifying algebraic expressions include:

• Not distributing the coefficient correctly.
• Not combining like terms correctly.
• Not simplifying the expression correctly.
• Not following the order of operations.

Q: How can I practice expanding and simplifying algebraic expressions?

A: You can practice expanding and simplifying algebraic expressions by working through examples and exercises in your textbook or online resources. You can also try creating your own examples and exercises to challenge yourself.

Q: What are some real-world applications of expanding and simplifying algebraic expressions?

A: Expanding and simplifying algebraic expressions has numerous real-world applications, including:

• Science and engineering: Algebraic expressions are used to model real-world phenomena, such as the motion of objects and the behavior of electrical circuits.
• Economics: Algebraic expressions are used to model economic systems, such as supply and demand curves.
• Computer science: Algebraic expressions are used to model algorithms and data structures.

Conclusion

Expanding and simplifying algebraic expressions is a fundamental skill in mathematics. By following the steps outlined in this article and practicing regularly, you can become proficient in expanding and simplifying algebraic expressions. Remember to distribute the coefficient to each term, combine like terms, and simplify the expression to get the final result.