Fill in the blanks. Then, choose the property of multiplication you used.

(a) $\square \times(6 \times 9)=(4 \times 6) \times 9$
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\hline (b) $2 \times 0=$ □ &

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\hline (c) □ $\times 9=9 \times 5$ &

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\hline (d) $1 \times \square=4$ &

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We are given four equations involving a placeholder $\square$ and need to identify the multiplication property used in each case. The properties we consider are the associative property, commutative property, identity property, and zero property of multiplication.

Solving part (a)

For the equation $\square \times (6 \times 9) = (4 \times 6) \times 9$, we use the associative property of multiplication, which states that $a \times (b \times c) = (a \times b) \times c$. Comparing $\square \times (6 \times 9)$ with $(4 \times 6) \times 9$, we recognize that this is a direct application where $a=4, b=6, c=9$. Therefore, the value of the placeholder is $\square = 4$.

Solving part (b)

For the equation $2 \times 0 = \square$, we apply the zero property of multiplication, which states that any number multiplied by zero is zero. Thus, $2 \times 0 = 0$, which gives the placeholder $\square = 0$.

Solving part (c)

For the equation $\square \times 9 = 9 \times 5$, we use the commutative property of multiplication, which states that $a \times b = b \times a$. Comparing $\square \times 9$ with $9 \times 5$, we can see that the order of the factors has been swapped. Thus, the value of the placeholder is $\square = 5$.

Solving part (d)

For the equation $1 \times \square = 4$, we use the identity property of multiplication, which states that any number multiplied by 1 remains unchanged, i.e., $1 \times a = a$. Therefore, for $1 \times \square = 4$, the value of the placeholder must be $\square = 4$.

Conclusion

Summarizing our findings:

(a) $\boxed{4}$, Associative Property
(b) $\boxed{0}$, Zero Property
(c) $\boxed{5}$, Commutative Property
(d) $\boxed{4}$, Identity Property

Examples

These properties of multiplication are fundamental in simplifying expressions and solving algebraic equations. For instance, the commutative property helps in rearranging terms in a long multiplication to make mental math easier (e.g., $2 \times 17 \times 5 = 2 \times 5 \times 17 = 10 \times 17 = 170$). Similarly, understanding the identity and zero properties allows for the quick evaluation of complex looking expressions that reduce down to 0 or leave the variable unchanged, which is crucial when building a foundation for more advanced algebra and calculus.

(a) $\square \times(6 \times 9)=(4 \times 6) \times 9$	\begin{tabular}[t]{l} □
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