Simplify The Expression:${ 8(6 + 8x) - 8x(7 + 4x) }$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of the underlying concepts. In this article, we will focus on simplifying the given expression: $8(6 + 8x) - 8x(7 + 4x)$. We will use various techniques, such as the distributive property and combining like terms, to simplify the expression.

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 the parentheses. In the given expression, we have two sets of parentheses: $6 + 8x$ and $7 + 4x$. We can 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 algebraic expressions.

Applying the Distributive Property

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

$8(6 + 8x) - 8x(7 + 4x)$

Using the distributive property, we can expand the first set of parentheses as follows:

$8(6 + 8x) = 8(6) + 8(8x)$

$= 48 + 64x$

Now, let's expand the second set of parentheses:

$8x(7 + 4x) = 8x(7) + 8x(4x)$

$= 56x + 32x^2$

Combining Like Terms

Now that we have expanded both sets of parentheses, we can combine like terms to simplify the expression. 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:

$a + a = 2a$

where $a$ is a like term.

Combining Like Terms

Let's combine like terms in the given expression:

$48 + 64x - 56x - 32x^2$

We can combine the like terms $64x$ and $-56x$ as follows:

$64x - 56x = 8x$

So, the expression becomes:

$48 + 8x - 32x^2$

Final Simplification

Now that we have combined like terms, we can simplify the expression further by rearranging the terms in descending order of the variable's exponent.

Final Simplification Formula

The formula for final simplification is:

$a + b + c = c + b + a$

where $a$, $b$, and $c$ are algebraic expressions.

Final Simplification

Let's simplify the expression by rearranging the terms in descending order of the variable's exponent:

$-32x^2 + 8x + 48$

Conclusion

In this article, we simplified the given expression: $8(6 + 8x) - 8x(7 + 4x)$. We used the distributive property to expand the expression and combined like terms to simplify it further. The final simplified expression is: $-32x^2 + 8x + 48$. This expression is now in its simplest form, and we can use it to solve various mathematical problems.

Frequently Asked Questions

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 parentheses.

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply each term inside the parentheses with the term outside the parentheses.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms.

Step-by-Step Solution

Step 1: Apply the Distributive Property

Apply the distributive property to the given expression:

$8(6 + 8x) - 8x(7 + 4x)$

$= 8(6) + 8(8x) - 8x(7) - 8x(4x)$

$= 48 + 64x - 56x - 32x^2$

Step 2: Combine Like Terms

Combine like terms in the expression:

$48 + 64x - 56x - 32x^2$

$= 48 + 8x - 32x^2$

Step 3: Final Simplification

Rearrange the terms in descending order of the variable's exponent:

$-32x^2 + 8x + 48$

The final answer is: $-32x^2 + 8x + 48$

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Introduction

In our previous article, we simplified the given expression: $8(6 + 8x) - 8x(7 + 4x)$. We used the distributive property to expand the expression and combined like terms to simplify it further. In this article, we will answer some frequently asked questions related to the simplification of the expression.

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 parentheses.

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply each term inside the parentheses with the term outside the parentheses.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms.

Q: What is the difference between the distributive property and the commutative property?

A: The distributive property allows us to expand expressions by multiplying each term inside the parentheses with the term outside the parentheses. The commutative property, on the other hand, allows us to rearrange the terms in an expression without changing its value.

Q: How do I simplify an expression with multiple sets of parentheses?

A: To simplify an expression with multiple sets of parentheses, you need to apply the distributive property to each set of parentheses separately and then combine like terms.

Q: What is the importance of simplifying expressions?

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

• Reduce the complexity of an expression
• Make it easier to solve mathematical problems
• Identify patterns and relationships between variables
• Make predictions and conclusions based on mathematical models

Q: How do I check if an expression is simplified?

A: To check if an expression is simplified, you need to:

• Look for any like terms that can be combined
• Check if the expression can be further simplified using the distributive property or other algebraic properties
• Verify that the expression is in its simplest form

Step-by-Step Solution

Step 1: Apply the Distributive Property

Apply the distributive property to the given expression:

$8(6 + 8x) - 8x(7 + 4x)$

$= 8(6) + 8(8x) - 8x(7) - 8x(4x)$

$= 48 + 64x - 56x - 32x^2$

Step 2: Combine Like Terms

Combine like terms in the expression:

$48 + 64x - 56x - 32x^2$

$= 48 + 8x - 32x^2$

Step 3: Final Simplification

Rearrange the terms in descending order of the variable's exponent:

$-32x^2 + 8x + 48$

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression: $8(6 + 8x) - 8x(7 + 4x)$. We also provided a step-by-step solution to simplify the expression. We hope that this article has helped you to understand the concept of simplifying expressions and how to apply it in mathematical problems.

Frequently Asked Questions

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 parentheses.

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to multiply each term inside the parentheses with the term outside the parentheses.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the like terms.

Step-by-Step Solution

Step 1: Apply the Distributive Property

Apply the distributive property to the given expression:

$8(6 + 8x) - 8x(7 + 4x)$

$= 8(6) + 8(8x) - 8x(7) - 8x(4x)$

$= 48 + 64x - 56x - 32x^2$

Step 2: Combine Like Terms

Combine like terms in the expression:

$48 + 64x - 56x - 32x^2$

$= 48 + 8x - 32x^2$

Step 3: Final Simplification

Rearrange the terms in descending order of the variable's exponent:

$-32x^2 + 8x + 48$

The final answer is: $-32x^2 + 8x + 48$