Which Choice Is Equivalent To The Expression Below? 4 7 − 3 X 7 − X 7 4 \sqrt{7} - 3x \sqrt{7} - X \sqrt{7} 4 7 ​ − 3 X 7 ​ − X 7 ​ A. 4 7 − 4 X 7 4 \sqrt{7} - 4x \sqrt{7} 4 7 ​ − 4 X 7 ​ B. − 2 X 7 -2x \sqrt{7} − 2 X 7 ​ C. − X 2 -x^2 − X 2 D. 0

Understanding the Problem

When dealing with algebraic expressions, it's essential to simplify them to make calculations easier and more manageable. In this article, we'll focus on simplifying the given expression: $4 \sqrt{7} - 3x \sqrt{7} - x \sqrt{7}$. Our goal is to find the equivalent expression among the given options.

Breaking Down the Expression

To simplify the given expression, we need to combine like terms. The expression contains three terms: $4 \sqrt{7}$, $-3x \sqrt{7}$, and $-x \sqrt{7}$. We can combine the terms with the same variable, which is $x$ in this case.

$4 \sqrt{7} - 3x \sqrt{7} - x \sqrt{7}$

Combining Like Terms

We can combine the terms with the variable $x$ by adding or subtracting their coefficients. In this case, we have:

$-3x \sqrt{7} - x \sqrt{7} = (-3 - 1)x \sqrt{7} = -4x \sqrt{7}$

Now, we can rewrite the original expression as:

$4 \sqrt{7} - 4x \sqrt{7}$

Evaluating the Options

We have simplified the expression to $4 \sqrt{7} - 4x \sqrt{7}$. Now, let's evaluate the given options to find the equivalent expression.

Option A: $4 \sqrt{7} - 4x \sqrt{7}$

This option is identical to the simplified expression we obtained earlier. Therefore, it is the correct answer.

Option B: $-2x \sqrt{7}$

This option is not equivalent to the simplified expression. The variable $x$ is squared in the original expression, but it is not squared in this option.

Option C: $-x^2$

This option is also not equivalent to the simplified expression. The variable $x$ is squared in this option, but it is not squared in the original expression.

Option D: 0

This option is not equivalent to the simplified expression. The original expression is not equal to zero.

Conclusion

In conclusion, the correct answer is Option A: $4 \sqrt{7} - 4x \sqrt{7}$. This option is equivalent to the simplified expression we obtained earlier. The other options are not equivalent to the simplified expression.

Tips and Tricks

When dealing with algebraic expressions, it's essential to simplify them to make calculations easier and more manageable. Here are some tips and tricks to help you simplify expressions:

• Combine like terms by adding or subtracting their coefficients.
• Use the distributive property to expand expressions.
• Use the commutative property to rearrange terms.
• Use the associative property to group terms.

By following these tips and tricks, you can simplify expressions and make calculations easier and more manageable.

Common Mistakes

When dealing with algebraic expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

• Failing to combine like terms.
• Failing to use the distributive property.
• Failing to use the commutative property.
• Failing to use the associative property.

By avoiding these common mistakes, you can simplify expressions and make calculations easier and more manageable.

Real-World Applications

Simplifying algebraic expressions has many real-world applications. Here are some examples:

• Simplifying expressions can help you solve equations and inequalities.
• Simplifying expressions can help you graph functions.
• Simplifying expressions can help you solve systems of equations.
• Simplifying expressions can help you solve optimization problems.

By simplifying expressions, you can make calculations easier and more manageable, and you can solve a wide range of problems in mathematics and other fields.

Final Thoughts

Frequently Asked Questions

In this article, we'll answer some frequently asked questions about simplifying algebraic expressions.

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that contains variables, constants, and mathematical operations. Examples of algebraic expressions include $2x + 3$, $x^2 - 4$, and $3x^2 + 2x - 1$.

Q: Why is it important to simplify algebraic expressions?

A: Simplifying algebraic expressions is important because it makes calculations easier and more manageable. By simplifying expressions, you can solve equations and inequalities, graph functions, and solve systems of equations.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, you need to combine like terms, use the distributive property, use the commutative property, and use the associative property. Here are some steps to follow:

1. Combine like terms by adding or subtracting their coefficients.
2. Use the distributive property to expand expressions.
3. Use the commutative property to rearrange terms.
4. Use the associative property to group terms.

Q: What are like terms?

A: Like terms are terms that have the same variable and exponent. For example, $2x$ and $3x$ are like terms because they have the same variable ($x$) and exponent (1).

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract their coefficients. For example, if you have the expression $2x + 3x$, you can combine the like terms by adding their coefficients: $2x + 3x = (2 + 3)x = 5x$.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that a single term can be distributed to multiple terms. For example, if you have the expression $2(x + 3)$, you can use the distributive property to expand it: $2(x + 3) = 2x + 6$.

Q: What is the commutative property?

A: The commutative property is a mathematical property that states that the order of terms does not change the result. For example, if you have the expression $2x + 3$, you can rearrange the terms using the commutative property: $2x + 3 = 3 + 2x$.

Q: What is the associative property?

A: The associative property is a mathematical property that states that the order in which you perform operations does not change the result. For example, if you have the expression $(2x + 3) + 4$, you can use the associative property to group the terms: $(2x + 3) + 4 = 2x + (3 + 4) = 2x + 7$.

Q: How do I know when to simplify an algebraic expression?

A: You should simplify an algebraic expression when you need to solve an equation or inequality, graph a function, or solve a system of equations. Simplifying expressions can help you make calculations easier and more manageable.

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

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

• Failing to combine like terms.
• Failing to use the distributive property.
• Failing to use the commutative property.
• Failing to use the associative property.

By avoiding these common mistakes, you can simplify expressions and make calculations easier and more manageable.

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by working through examples and exercises. You can also use online resources, such as math websites and apps, to practice simplifying expressions.

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics and other fields. By following the tips and tricks outlined in this article, you can simplify expressions and make calculations easier and more manageable. Remember to combine like terms, use the distributive property, use the commutative property, and use the associative property to simplify expressions. By avoiding common mistakes and practicing simplifying expressions, you can become proficient in simplifying algebraic expressions.