Question

complete the statements about the associative property and equivalent expressions. the associative property allows us to change the grouping of the factors that are multiplied together to create dropdown. when changing the grouping, the order of the factors stays the same; only the dropdown change position. the expression dropdown is an equivalent expression to w(xy)z. the expression dropdown is also an equivalent expression to w(xy)z.

Answer:

To solve this, we recall the associative property of multiplication: \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \), which means grouping can change but order (for associative; commutative is order) and the result (equivalent expression) remains. Let's analyze each part:

1. First Blank:

The associative property of multiplication (since we’re multiplying factors) states that changing the grouping of factors being multiplied does not change the product (or creates an equivalent expression). So the first blank is about creating an equivalent expression (or a product with the same value).

2. Second Blank:

When using the associative property, the order of the factors stays the same—only the grouping symbols (parentheses/brackets) change position. The commutative property changes order, but associative changes grouping.

3. & 4. Equivalent to \( w(xy)z \):

Using the associative property, we can regroup the factors. Let’s apply \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \):

• Regroup \( w \) and \( (xy) \) first: \( (w(xy))z \) (or \( (wx y) z \), but more precisely, \( (w \cdot (xy)) \cdot z \) can be regrouped as \( w \cdot ((xy) \cdot z) \) or \( (w \cdot x \cdot y) \cdot z \), or \( w \cdot x \cdot y \cdot z \) (since multiplication is associative and commutative). Wait, let's clarify:

The original expression is \( w(xy)z \). Let’s rewrite with parentheses for grouping:

• Option 1: Regroup \( w \) and \( xy \) first, then multiply by \( z \): \( (w(xy))z \) (but this is the same as original, so maybe another grouping).
• Option 2: Regroup \( (xy) \) and \( z \) first: \( w((xy)z) \) (since \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \), so \( w(xy)z = w \cdot (xy) \cdot z = w \cdot ((xy) \cdot z) \)).
• Option 3: Regroup \( w \) and \( x \) (but no, associative is about grouping, not order). Wait, actually, \( w(xy)z = (w \cdot x \cdot y) \cdot z = w \cdot (x \cdot y \cdot z) = (w \cdot z) \cdot (x \cdot y) \), etc. But the key is that equivalent expressions use the associative property to regroup.

Filling the Blanks (Typical Options, Assuming Dropdowns):

1. The associative property allows us to change the grouping of the factors that are multiplied together to create an equivalent expression (or “the same product”).
2. When changing the grouping, the order of the factors stays the same; only the grouping symbols (parentheses) change position.
3. An equivalent expression to \( w(xy)z \) is \( (w x y) z \) (or \( w((xy)z) \), or \( (w z)(xy) \), etc.—depends on dropdown options).
4. Another equivalent expression could be \( w x y z \) (since multiplication is associative, we can remove all grouping: \( w \cdot x \cdot y \cdot z \)) or \( (w)(xy z) \).

Example Dropdown Answers (Common Scenarios):

• First blank: an equivalent expression
• Second blank: grouping symbols (or “parentheses”)
• Third blank: \( (w(xy))z \) (or \( w((xy)z) \), or \( (w x y) z \))
• Fourth blank: \( w x y z \) (or \( (w z)(xy) \), etc.)

If we assume the dropdowns have options like:

• First: an equivalent expression
• Second: grouping symbols
• Third: \( (w x y) z \)
• Fourth: \( w x y z \)

Final Answer (Filled Blanks):

1. The associative property allows us to change the grouping of the factors that are multiplied together to create \(\boldsymbol{\text{an equivalent expression}}\).
2. When changing the grouping, the order of the factors stays the same; only the \(\boldsymbol{\text{grouping symbols}}\) change position.
3. The expression \(\boldsymbol{(wxy)z}\) (or similar) is an equivalent expression to \( w(xy)z \).
4. The expression \(\boldsymbol{wxyz}\) (or similar) is also an equivalent expression to \( w(xy)z \).

(Note: Exact wording depends on dropdown options, but the logic is based on the associative property of multiplication.)