Simplify The Expression:${ 2(2x + 1) + 8(x + 4) }$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. In this article, we will focus on simplifying the given expression: 2(2x + 1) + 8(x + 4). We will break down the expression step by step, using the distributive property and combining like terms.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside. In the given expression, we have two sets of parentheses: 2(2x + 1) and 8(x + 4). We will use the distributive property to expand each set of parentheses.

Distributive Property Formula

The distributive property formula is:

a(b + c) = ab + ac

where a, b, and c are variables or constants.

Applying the Distributive Property

Let's apply the distributive property to the given expression:

2(2x + 1) = 2(2x) + 2(1) = 4x + 2

8(x + 4) = 8(x) + 8(4) = 8x + 32

Combining Like Terms

Now that we have expanded each set of parentheses, we can combine like terms. Like terms are terms that have the same variable raised to the same power.

Combining Like Terms Formula

The formula for combining like terms is:

a + a = 2a a + b = a + b (if a and b are not like terms)

Combining Like Terms in the Expression

Let's combine like terms in the expression:

4x + 2 + 8x + 32 = (4x + 8x) + (2 + 32) = 12x + 34

Conclusion

In this article, we simplified the expression 2(2x + 1) + 8(x + 4) using the distributive property and combining like terms. We broke down the expression step by step, expanding each set of parentheses and combining like terms. The final simplified expression is 12x + 34.

Tips and Tricks

  • When simplifying expressions, always start by applying the distributive property to expand each set of parentheses.
  • Combine like terms by adding or subtracting the coefficients of the like terms.
  • Make sure to check your work by plugging in a value for the variable and simplifying the expression.

Real-World Applications

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

  • Solving systems of equations
  • Finding the maximum or minimum value of a function
  • Modeling real-world situations using algebraic expressions

Practice Problems

Try simplifying the following expressions using the distributive property and combining like terms:

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

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. By following the steps outlined in this article, you can simplify expressions like 2(2x + 1) + 8(x + 4) and become more confident in your ability to solve algebraic problems.

Introduction

In our previous article, we simplified the expression 2(2x + 1) + 8(x + 4) using the distributive property and combining like terms. In this article, we will answer some frequently asked questions about simplifying algebraic expressions.

Q&A

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by multiplying each term inside the parentheses with the term outside. The formula for the distributive property is:

a(b + c) = ab + ac

where a, b, and c are variables or constants.

Q: How do I apply the distributive property?

A: To apply the distributive property, simply multiply each term inside the parentheses with the term outside. For example, in the expression 2(2x + 1), we would multiply 2 with each term inside the parentheses: 2(2x) + 2(1).

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, 2x and 4x 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, simply add or subtract the coefficients of the like terms. For example, in the expression 4x + 8x, we would combine the like terms by adding the coefficients: 4x + 8x = 12x.

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

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

  • Forgetting to apply the distributive property
  • Not combining like terms
  • Making errors when multiplying or dividing terms

Q: How do I check my work when simplifying expressions?

A: To check your work, simply plug in a value for the variable and simplify the expression. For example, if we plug in x = 1 into the expression 2(2x + 1) + 8(x + 4), we get:

2(2(1) + 1) + 8(1 + 4) = 2(3) + 8(5) = 6 + 40 = 46

This confirms that our simplified expression is correct.

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

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

  • Solving systems of equations
  • Finding the maximum or minimum value of a function
  • Modeling real-world situations using algebraic expressions

Tips and Tricks

  • Always start by applying the distributive property to expand each set of parentheses.
  • Combine like terms by adding or subtracting the coefficients of the like terms.
  • Make sure to check your work by plugging in a value for the variable and simplifying the expression.

Practice Problems

Try simplifying the following expressions using the distributive property and combining like terms:

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

Final Thoughts

Simplifying algebraic expressions is a crucial skill in mathematics, and it's essential to understand the rules and techniques involved. By following the steps outlined in this article, you can simplify expressions like 2(2x + 1) + 8(x + 4) and become more confident in your ability to solve algebraic problems.