Expand And Simplify $8(5c - 2) + 7$.

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 expression $8(5c - 2) + 7$. We will break down the process into manageable steps, making it easy to understand and follow along.

Understanding the Expression

Before we begin, let's take a closer look at the expression $8(5c - 2) + 7$. This expression consists of two main parts: the first part is the product of $8$ and the expression $5c - 2$, and the second part is the constant term $7$. Our goal is to simplify this expression by combining like terms and eliminating any unnecessary parentheses.

Step 1: Distribute the 8

To expand the expression, we need to distribute the $8$ to both terms inside the parentheses. This means we will multiply the $8$ by the $5c$ and the $-2$ separately.

8(5c - 2) = 8(5c) - 8(2)

Step 2: Simplify the Terms

Now that we have distributed the $8$, we can simplify the terms by multiplying the numbers.

8(5c) = 40c
-8(2) = -16

Step 3: Combine Like Terms

The expression now looks like this: $40c - 16 + 7$. We can combine the like terms by adding or subtracting the coefficients of the variables.

40c - 16 + 7 = 40c - 9

Step 4: Final Simplification

The final simplified expression is $40c - 9$. This is the expanded and simplified form of the original expression $8(5c - 2) + 7$.

Conclusion

Expanding and simplifying algebraic expressions is an essential skill in mathematics. By following the steps outlined in this article, you can master the process of expanding and simplifying expressions like $8(5c - 2) + 7$. Remember to distribute the terms, simplify the expressions, combine like terms, and finally, simplify the expression to its simplest form.

Tips and Tricks

• Always follow the order of operations (PEMDAS) when simplifying expressions.
• Use parentheses to group terms and make it easier to simplify the expression.
• Combine like terms by adding or subtracting the coefficients of the variables.
• Simplify the expression by eliminating any unnecessary parentheses.

Common Mistakes

• Failing to distribute the terms correctly.
• Not simplifying the expressions properly.
• Not combining like terms correctly.
• Not eliminating unnecessary parentheses.

Real-World Applications

Expanding and simplifying algebraic expressions has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, you may need to simplify expressions to describe the motion of objects, while in engineering, you may need to simplify expressions to design and optimize systems.

Practice Problems

Try simplifying the following expressions:

• $3(2x + 5) + 2$
• $4(x - 2) + 1$
• $2(3x - 1) + 5$

Answer Key

• $3(2x + 5) + 2 = 6x + 17$
• $4(x - 2) + 1 = 4x - 7$
• $2(3x - 1) + 5 = 6x + 3$

Conclusion

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Q: What is the first step in expanding and simplifying an algebraic expression?

A: The first step in expanding and simplifying an algebraic expression is to distribute the terms inside the parentheses. This means multiplying the number outside the parentheses by each term inside the parentheses.

Q: How do I simplify an expression with multiple terms inside the parentheses?

A: To simplify an expression with multiple terms inside the parentheses, you need to distribute the terms outside the parentheses to each term inside the parentheses. For example, if you have the expression $3(2x + 5 + 2)$, you would distribute the $3$ to each term inside the parentheses: $3(2x + 5 + 2) = 3(2x) + 3(5) + 3(2)$.

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

A: Expanding an expression means multiplying the terms inside the parentheses by the number outside the parentheses, while simplifying an expression means combining like terms and eliminating any unnecessary parentheses.

Q: How do I combine like terms in an expression?

A: To combine like terms in an expression, you need to add or subtract the coefficients of the variables. For example, if you have the expression $2x + 3x$, you would combine the like terms by adding the coefficients: $2x + 3x = 5x$.

Q: What is the order of operations (PEMDAS) and how does it apply to expanding and simplifying expressions?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when simplifying an expression. The acronym PEMDAS stands for:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: Can you provide an example of how to apply the order of operations to an expression?

A: Let's consider the expression $3(2x + 5) + 2$. To simplify this expression, we would follow the order of operations as follows:

1. Evaluate the expression inside the parentheses: $2x + 5$
2. Multiply the number outside the parentheses by the expression inside the parentheses: $3(2x + 5) = 6x + 15$
3. Add the constant term: $6x + 15 + 2 = 6x + 17$

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

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

• Failing to distribute the terms correctly
• Not simplifying the expressions properly
• Not combining like terms correctly
• Not eliminating unnecessary parentheses

Q: How can I practice expanding and simplifying expressions?

A: You can practice expanding and simplifying expressions by working through exercises and problems in a textbook or online resource. You can also try simplifying expressions on your own and then checking your work with a calculator or online tool.

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

A: Expanding and simplifying expressions has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, you may need to simplify expressions to describe the motion of objects, while in engineering, you may need to simplify expressions to design and optimize systems.

Conclusion

Expanding and simplifying algebraic expressions is a crucial skill in mathematics. By following the steps outlined in this article and practicing regularly, you can master the process of expanding and simplifying expressions. Remember to distribute the terms, simplify the expressions, combine like terms, and finally, simplify the expression to its simplest form.