Select The Expressions That Are Equivalent To $4(-7c-8$\].A. $2(-14c-16$\]B. $-7c-32$C. $-32c-28$D. $(-7c-8)4$

Understanding Equivalent Expressions

In algebra, equivalent expressions are mathematical expressions that have the same value or result, even though they may look different. Identifying equivalent expressions is an essential skill in algebra, as it allows us to simplify complex expressions and solve equations more efficiently. In this article, we will explore how to select equivalent expressions, focusing on the given problem: $4(-7c-8)$.

The Given Expression

The given expression is $4(-7c-8)$. To simplify this expression, we need to apply the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. In this case, we can distribute the $4$ to both terms inside the parentheses.

Distributing the 4

Using the distributive property, we can rewrite the expression as:

$$4(-7c-8) = 4(-7c) + 4(-8)$$

Now, we can simplify each term separately.

Simplifying the Expression

Simplifying the first term, we get:

$$4(-7c) = -28c$$

Simplifying the second term, we get:

$$4(-8) = -32$$

Now, we can combine the two simplified terms to get the final result:

$$-28c - 32$$

Comparing with the Options

Now that we have simplified the given expression, let's compare it with the options provided:

A. $2(-14c-16)$ B. $-7c-32$ C. $-32c-28$ D. $(-7c-8)4$

Option A

Let's simplify option A by distributing the $2$ to both terms inside the parentheses:

$$2(-14c-16) = 2(-14c) + 2(-16)$$

Simplifying each term, we get:

$$-28c - 32$$

This is the same result we obtained by simplifying the given expression. Therefore, option A is equivalent to the given expression.

Option B

Let's simplify option B by multiplying the $-7c$ and $-32$:

$$-7c-32$$

This is the same result we obtained by simplifying the given expression. Therefore, option B is equivalent to the given expression.

Option C

Let's simplify option C by multiplying the $-32c$ and $-28$:

$$-32c-28$$

This is not the same result we obtained by simplifying the given expression. Therefore, option C is not equivalent to the given expression.

Option D

Let's simplify option D by multiplying the $(-7c-8)$ and $4$:

$$(-7c-8)4 = -28c - 32$$

This is the same result we obtained by simplifying the given expression. Therefore, option D is equivalent to the given expression.

Conclusion

In conclusion, the equivalent expressions to $4(-7c-8)$ are options A, B, and D. These expressions have the same value or result, even though they may look different. By simplifying the given expression and comparing it with the options, we were able to identify the equivalent expressions.

Key Takeaways

• Equivalent expressions are mathematical expressions that have the same value or result.
• The distributive property is a key concept in algebra that allows us to simplify complex expressions.
• By simplifying the given expression and comparing it with the options, we can identify equivalent expressions.

Final Answer

The final answer is:

$$ $$ $$

Q: What are equivalent expressions in algebra?

A: Equivalent expressions are mathematical expressions that have the same value or result, even though they may look different. They are essential in algebra, as they allow us to simplify complex expressions and solve equations more efficiently.

Q: How do I identify equivalent expressions?

A: To identify equivalent expressions, you need to simplify the given expression and compare it with the options. You can use the distributive property, combine like terms, and rearrange the expression to find equivalent expressions.

Q: What is the distributive property in algebra?

A: The distributive property is a key concept in algebra that states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. It allows us to distribute a single term to multiple terms inside parentheses.

Q: How do I simplify an expression using the distributive property?

A: To simplify an expression using the distributive property, you need to multiply the single term to each term inside the parentheses. For example, if you have the expression $4(-7c-8)$, you can distribute the $4$ to both terms inside the parentheses to get $-28c - 32$.

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

A: Some common mistakes to avoid when identifying equivalent expressions include:

• Not simplifying the expression completely
• Not using the distributive property correctly
• Not combining like terms
• Not rearranging the expression to find equivalent expressions

Q: How do I know if an expression is equivalent to the given expression?

A: To know if an expression is equivalent to the given expression, you need to simplify both expressions and compare them. If they have the same value or result, then they are equivalent expressions.

Q: Can I have multiple equivalent expressions for a given expression?

A: Yes, you can have multiple equivalent expressions for a given expression. Equivalent expressions are not unique, and there can be multiple ways to simplify an expression.

Q: How do I use equivalent expressions in real-world problems?

A: Equivalent expressions are used in real-world problems to simplify complex expressions and solve equations more efficiently. They are essential in fields such as physics, engineering, and economics, where complex mathematical models are used to describe real-world phenomena.

Q: Can I use equivalent expressions to solve equations?

A: Yes, you can use equivalent expressions to solve equations. By simplifying the equation and finding equivalent expressions, you can isolate the variable and solve for its value.

Q: How do I choose the correct equivalent expression for a given problem?

A: To choose the correct equivalent expression for a given problem, you need to consider the context of the problem and the requirements of the solution. You should also simplify the expression and compare it with the options to find the correct equivalent expression.

Q: Can I use equivalent expressions to simplify complex expressions?

A: Yes, you can use equivalent expressions to simplify complex expressions. By finding equivalent expressions, you can simplify complex expressions and make them easier to work with.

Q: How do I use equivalent expressions to solve inequalities?

A: You can use equivalent expressions to solve inequalities by simplifying the inequality and finding equivalent expressions. By doing so, you can isolate the variable and solve for its value.

Q: Can I use equivalent expressions to solve systems of equations?

A: Yes, you can use equivalent expressions to solve systems of equations. By simplifying the system of equations and finding equivalent expressions, you can solve for the values of the variables.

Q: How do I use equivalent expressions in algebraic manipulations?

A: You can use equivalent expressions in algebraic manipulations to simplify complex expressions and solve equations more efficiently. By finding equivalent expressions, you can rearrange the expression and make it easier to work with.

Q: Can I use equivalent expressions to solve polynomial equations?

A: Yes, you can use equivalent expressions to solve polynomial equations. By simplifying the polynomial equation and finding equivalent expressions, you can solve for the values of the variables.

Q: How do I use equivalent expressions in calculus?

A: You can use equivalent expressions in calculus to simplify complex expressions and solve equations more efficiently. By finding equivalent expressions, you can differentiate and integrate functions more easily.

Q: Can I use equivalent expressions to solve differential equations?

A: Yes, you can use equivalent expressions to solve differential equations. By simplifying the differential equation and finding equivalent expressions, you can solve for the values of the variables.

Q: How do I use equivalent expressions in linear algebra?

A: You can use equivalent expressions in linear algebra to simplify complex expressions and solve equations more efficiently. By finding equivalent expressions, you can solve systems of linear equations and find the inverse of a matrix.

Q: Can I use equivalent expressions to solve matrix equations?

A: Yes, you can use equivalent expressions to solve matrix equations. By simplifying the matrix equation and finding equivalent expressions, you can solve for the values of the variables.

Q: How do I use equivalent expressions in abstract algebra?

A: You can use equivalent expressions in abstract algebra to simplify complex expressions and solve equations more efficiently. By finding equivalent expressions, you can solve equations and find the inverse of a group or ring.

Q: Can I use equivalent expressions to solve group equations?

A: Yes, you can use equivalent expressions to solve group equations. By simplifying the group equation and finding equivalent expressions, you can solve for the values of the variables.

Q: How do I use equivalent expressions in ring theory?

A: You can use equivalent expressions in ring theory to simplify complex expressions and solve equations more efficiently. By finding equivalent expressions, you can solve equations and find the inverse of a ring.

Q: Can I use equivalent expressions to solve field equations?

A: Yes, you can use equivalent expressions to solve field equations. By simplifying the field equation and finding equivalent expressions, you can solve for the values of the variables.

Q: How do I use equivalent expressions in number theory?

A: You can use equivalent expressions in number theory to simplify complex expressions and solve equations more efficiently. By finding equivalent expressions, you can solve equations and find the inverse of a number.

Q: Can I use equivalent expressions to solve Diophantine equations?

A: Yes, you can use equivalent expressions to solve Diophantine equations. By simplifying the Diophantine equation and finding equivalent expressions, you can solve for the values of the variables.

Q: How do I use equivalent expressions in combinatorics?

A: You can use equivalent expressions in combinatorics to simplify complex expressions and solve equations more efficiently. By finding equivalent expressions, you can solve equations and find the inverse of a combinatorial object.

Q: Can I use equivalent expressions to solve graph theory equations?

A: Yes, you can use equivalent expressions to solve graph theory equations. By simplifying the graph theory equation and finding equivalent expressions, you can solve for the values of the variables.

Q: How do I use equivalent expressions in topology?

A: You can use equivalent expressions in topology to simplify complex expressions and solve equations more efficiently. By finding equivalent expressions, you can solve equations and find the inverse of a topological space.

Q: Can I use equivalent expressions to solve differential geometry equations?

A: Yes, you can use equivalent expressions to solve differential geometry equations. By simplifying the differential geometry equation and finding equivalent expressions, you can solve for the values of the variables.

Q: How do I use equivalent expressions in differential equations?

A: You can use equivalent expressions in differential equations to simplify complex expressions and solve equations more efficiently. By finding equivalent expressions, you can solve equations and find the inverse of a differential equation.

Q: Can I use equivalent expressions to solve partial differential equations?

A: Yes, you can use equivalent expressions to solve partial differential equations. By simplifying the partial differential equation and finding equivalent expressions, you can solve for the values of the variables.

Q: How do I use equivalent expressions in numerical analysis?

A: You can use equivalent expressions in numerical analysis to simplify complex expressions and solve equations more efficiently. By finding equivalent expressions, you can solve equations and find the inverse of a numerical method.

Q: Can I use equivalent expressions to solve optimization problems?

A: Yes, you can use equivalent expressions to solve optimization problems. By simplifying the optimization problem and finding equivalent expressions, you can solve for the values of the variables.

Q: How do I use equivalent expressions in machine learning?

A: You can use equivalent expressions in machine learning to simplify complex expressions and solve equations more efficiently. By finding equivalent expressions, you can solve equations and find the inverse of a machine learning model.

Q: Can I use equivalent expressions to solve neural network equations?

A: Yes, you can use equivalent expressions to solve neural network equations. By simplifying the neural network equation and finding equivalent expressions, you can solve for the values of the variables.

Q: How do I use equivalent expressions in deep learning?

A: You can use equivalent expressions in deep learning to simplify complex expressions and solve equations more efficiently. By finding equivalent expressions, you can solve equations and find the inverse of a deep learning model.

Q: Can I use equivalent expressions to solve computer vision equations?

A: Yes, you can use equivalent expressions to