Manipulating Algebraic Expressions

Oct 13, 2025 by windailys.com 128 views

Which Expression is Equivalent to ${$ (xy)z$}$?

Understanding the Problem

When dealing with algebraic expressions, it's essential to understand the order of operations and how to manipulate them to simplify or find equivalent expressions. In this case, we're given the expression ${$ (xy)z$}$ and asked to determine which of the provided options is equivalent to it.

The Distributive Property

To approach this problem, we need to recall the distributive property of multiplication over addition. This property states that for any numbers a, b, and c, the following equation holds:

a(b + c) = ab + ac

We can use this property to expand the given expression ${$ (xy)z$}$ and see if it matches any of the options.

Expanding the Expression

Using the distributive property, we can expand the expression ${$ (xy)z$}$ as follows:

${$ (xy)z$}$ = ${$ x(yz)$}$

This is because we can distribute the z to both the x and the y, resulting in x multiplied by the product of y and z.

Comparing with Options

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

A. ${$ (x+y)+z$}$ - This option is not equivalent to the expanded expression, as it involves adding x and y before multiplying by z.

B. ${ $2z(xy)\$}$ - This option is not equivalent to the expanded expression, as it involves multiplying z by the product of x and y, but also includes a factor of 2.

C. ${$ x(yz)$}$ - This option is equivalent to the expanded expression, as it matches the result of distributing z to both the x and the y.

D. ${$ x(y+z)$}$ - This option is not equivalent to the expanded expression, as it involves adding y and z before multiplying by x.

Conclusion

Based on the distributive property and the expansion of the given expression, we can conclude that the correct answer is:

C. ${$ x(yz)$}$

This option is equivalent to the expanded expression, and it correctly represents the result of distributing z to both the x and the y.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to expand expressions by distributing a factor to multiple terms. This property is essential for simplifying expressions, solving equations, and manipulating algebraic expressions.

Real-World Applications

The distributive property has numerous real-world applications, including:

Simplifying Expressions: The distributive property can be used to simplify complex expressions by expanding them and combining like terms.
Solving Equations: The distributive property can be used to solve equations by expanding and simplifying expressions.
Manipulating Algebraic Expressions: The distributive property can be used to manipulate algebraic expressions by expanding and simplifying them.

Common Mistakes

When working with the distributive property, it's essential to avoid common mistakes, including:

Forgetting to distribute: Failing to distribute a factor to multiple terms can lead to incorrect results.
Distributing incorrectly: Distributing a factor to the wrong terms can lead to incorrect results.
Not combining like terms: Failing to combine like terms after distributing a factor can lead to incorrect results.

Tips and Tricks

To master the distributive property, follow these tips and tricks:

Practice, practice, practice: The more you practice, the more comfortable you'll become with the distributive property.
Use visual aids: Visual aids, such as diagrams and charts, can help you understand the distributive property and how to apply it.
Break down complex expressions: Breaking down complex expressions into smaller, more manageable parts can help you apply the distributive property more effectively.

Conclusion

In conclusion, the distributive property is a fundamental concept in algebra that allows us to expand expressions by distributing a factor to multiple terms. By understanding the distributive property and how to apply it, you can simplify complex expressions, solve equations, and manipulate algebraic expressions with ease. Remember to practice, use visual aids, and break down complex expressions to master the distributive property.
Frequently Asked Questions (FAQs) About the Distributive Property

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that allows us to expand expressions by distributing a factor to multiple terms. It states that for any numbers a, b, and c, the following equation holds:

a(b + c) = ab + ac

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to distribute a factor to multiple terms. For example, if you have the expression 2(x + y), you can distribute the 2 to both the x and the y by multiplying each term by 2, resulting in 2x + 2y.

Q: What are some common mistakes to avoid when using the distributive property?

A: Some common mistakes to avoid when using the distributive property include:

Forgetting to distribute a factor to multiple terms
Distributing a factor to the wrong terms
Not combining like terms after distributing a factor

Q: How do I simplify expressions using the distributive property?

A: To simplify expressions using the distributive property, you need to expand the expression by distributing a factor to multiple terms and then combine like terms. For example, if you have the expression 2(x + y), you can simplify it by distributing the 2 to both the x and the y, resulting in 2x + 2y.

Q: Can I use the distributive property to solve equations?

A: Yes, you can use the distributive property to solve equations. By expanding and simplifying expressions using the distributive property, you can isolate the variable and solve for its value.

Q: What are some real-world applications of the distributive property?

A: Some real-world applications of the distributive property include:

Simplifying complex expressions in algebra and calculus
Solving equations in physics and engineering
Manipulating algebraic expressions in computer science and cryptography

Q: How do I know when to use the distributive property?

A: You should use the distributive property when you have an expression with multiple terms and a factor that can be distributed to each term. For example, if you have the expression 2(x + y), you can use the distributive property to distribute the 2 to both the x and the y.

Q: Can I use the distributive property with fractions?

A: Yes, you can use the distributive property with fractions. By distributing a factor to multiple terms, you can simplify expressions with fractions and solve equations.

Q: What are some tips and tricks for mastering the distributive property?

A: Some tips and tricks for mastering the distributive property include:

Practicing, practicing, practicing
Using visual aids, such as diagrams and charts
Breaking down complex expressions into smaller, more manageable parts

Q: How do I know if I'm using the distributive property correctly?

A: To know if you're using the distributive property correctly, you should:

Check that you're distributing the factor to the correct terms
Make sure you're combining like terms after distributing the factor
Verify that your answer is correct by checking it with the original expression

Conclusion