Which Expression Is Equivalent To $(xy)z$?A. $(x + Y) + Z$B. \$2z(xy)$[/tex\]C. $x(yz)$D. $x(y + Z)$

by ADMIN 110 views

Understanding the Problem

When dealing with algebraic expressions, it's essential to understand the order of operations and how to simplify complex expressions. In this problem, we're given the expression (xy)z and asked to find an equivalent expression from the given options.

The Order of Operations

To simplify the expression (xy)z, we need to follow the order of operations, which is a set of rules that dictate the order in which we perform mathematical operations. The order of operations is often remembered using the acronym PEMDAS, which stands for:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Applying the Order of Operations

Now, let's apply the order of operations to the expression (xy)z:

  1. Evaluate the expression inside the parentheses: xy
  2. Multiply the result by z: (xy)z = xy * z

Simplifying the Expression

To simplify the expression xy * z, we can use the associative property of multiplication, which states that the order in which we multiply numbers does not change the result. Using this property, we can rewrite the expression as:

xy * z = x * (y * z)

Evaluating the Options

Now, let's evaluate the given options to see which one is equivalent to the simplified expression x * (y * z):

Option A: (x + y) + z

This option is not equivalent to the simplified expression x * (y * z) because it involves addition, not multiplication.

Option B: 2z(xy)

This option is not equivalent to the simplified expression x * (y * z) because it involves multiplication by 2, not just multiplication.

Option C: x(yz)

This option is equivalent to the simplified expression x * (y * z) because it uses the associative property of multiplication to rewrite the expression.

Option D: x(y + z)

This option is not equivalent to the simplified expression x * (y * z) because it involves addition, not multiplication.

Conclusion

Based on the order of operations and the associative property of multiplication, we can conclude that the expression equivalent to (xy)z is option C: x(yz).

Final Answer

The final answer is option C: x(yz).

Understanding Equivalent Expressions

Equivalent expressions are algebraic expressions that have the same value, even if they are written differently. In the previous article, we discussed how to simplify the expression (xy)z and find an equivalent expression from the given options. In this article, we'll answer some frequently asked questions about equivalent expressions.

Q: What is the difference between equivalent expressions and identical expressions?

A: Equivalent expressions are algebraic expressions that have the same value, but may be written differently. Identical expressions, on the other hand, are algebraic expressions that are written exactly the same way.

Q: How do I determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, you can use the following steps:

  1. Simplify both expressions using the order of operations.
  2. Compare the simplified expressions to see if they are the same.
  3. If the simplified expressions are the same, then the original expressions are equivalent.

Q: Can equivalent expressions have different variables?

A: Yes, equivalent expressions can have different variables. For example, the expressions 2x and 4y are equivalent, even though they have different variables.

Q: Can equivalent expressions have different coefficients?

A: Yes, equivalent expressions can have different coefficients. For example, the expressions 3x and 6y are equivalent, even though they have different coefficients.

Q: How do I simplify complex expressions?

A: To simplify complex expressions, you can use the following steps:

  1. Use the order of operations to simplify the expression.
  2. Combine like terms.
  3. Use the associative property of multiplication to rewrite the expression.
  4. Use the distributive property of multiplication to rewrite the expression.

Q: What is the distributive property of multiplication?

A: The distributive property of multiplication states that a single term can be distributed to multiple terms. For example, the expression 2(x + y) can be rewritten as 2x + 2y using the distributive property.

Q: How do I use the distributive property of multiplication?

A: To use the distributive property of multiplication, you can follow these steps:

  1. Identify the single term that is being distributed.
  2. Identify the multiple terms that the single term is being distributed to.
  3. Rewrite the expression using the distributive property.

Q: What is the associative property of multiplication?

A: The associative property of multiplication states that the order in which we multiply numbers does not change the result. For example, the expression (x * y) * z can be rewritten as x * (y * z) using the associative property.

Q: How do I use the associative property of multiplication?

A: To use the associative property of multiplication, you can follow these steps:

  1. Identify the expression that is being rewritten.
  2. Rewrite the expression using the associative property.

Q: Can equivalent expressions have different exponents?

A: Yes, equivalent expressions can have different exponents. For example, the expressions x^2 and (x * x) are equivalent, even though they have different exponents.

Q: How do I simplify expressions with exponents?

A: To simplify expressions with exponents, you can use the following steps:

  1. Use the order of operations to simplify the expression.
  2. Combine like terms.
  3. Use the associative property of multiplication to rewrite the expression.
  4. Use the distributive property of multiplication to rewrite the expression.

Q: What is the order of operations?

A: The order of operations is a set of rules that dictate the order in which we perform mathematical operations. The order of operations is often remembered using the acronym PEMDAS, which stands for:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I use the order of operations?

A: To use the order of operations, you can follow these steps:

  1. Identify the expression that needs to be simplified.
  2. Evaluate any expressions inside parentheses first.
  3. Evaluate any exponential expressions next.
  4. Evaluate any multiplication and division operations from left to right.
  5. Finally, evaluate any addition and subtraction operations from left to right.

Q: Can equivalent expressions have different constants?

A: Yes, equivalent expressions can have different constants. For example, the expressions 2x and 4y are equivalent, even though they have different constants.

Q: How do I simplify expressions with constants?

A: To simplify expressions with constants, you can use the following steps:

  1. Use the order of operations to simplify the expression.
  2. Combine like terms.
  3. Use the associative property of multiplication to rewrite the expression.
  4. Use the distributive property of multiplication to rewrite the expression.

Q: What is the difference between equivalent expressions and equivalent equations?

A: Equivalent expressions are algebraic expressions that have the same value, while equivalent equations are algebraic equations that have the same solution.