Answered

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




(a)		$7 \times 0=0$
\begin{tabular}[t]{l}
□

Zero Property
\\

\hline (b) & $1 \times 8=8$ &


□

Identity Property
\\

\hline (c) □ $\times(2 \times 8)=(3 \times 2) \times 8$ & \multicolumn{2}{|r|}{

□

Associative Property
} \\

\hline (d) & $9 \times 8=8 \times 9$ &


□

Commutative Property
\\

\hline

\end{tabular}

Verified Answer :

  • The Associative Property of multiplication states that \((a \times b) \times c = a \times (b \times c)\).
  • We compare the given equation \(\boxed{\phantom{0}} \times (2 \times 8) = (3 \times 2) \times 8\) to this property.
  • By setting \(a=3\), \(b=2\), and \(c=8\), we see the left side must be \(3 \times (2 \times 8)\).
  • The missing number is \(\boxed{3}\).

Explanation

  1. Analyzing the Problem and Associative Property

The problem requires us to identify the missing number in the multiplication equation \(\boxed{\phantom{0}} \times (2 \times 8) = (3 \times 2) \times 8\) based on the Associative Property of multiplication. The Associative Property states that the way factors are grouped in a multiplication problem does not change the product. Mathematically, for any three numbers \(a\), \(b\), and \(c\), this is represented as \((a \times b) \times c = a \times (b \times c)\).

  1. Applying the Associative Property

We are given the equation \(\boxed{\phantom{0}} \times (2 \times 8) = (3 \times 2) \times 8\). According to the Associative Property, if we have an expression of the form \((a \times b) \times c\), it is equivalent to \(a \times (b \times c)\). Comparing our given equation to this structure, we have the right side as \((3 \times 2) \times 8\), where \(a=3\), \(b=2\), and \(c=8\). Therefore, the left side must be \(3 \times (2 \times 8)\) to satisfy the property. This reveals that the missing number in the blank must be 3.

Examples

The Associative Property is incredibly useful in daily life when performing mental math. For example, if you need to calculate the total cost of 3 packs of pens, where each pack contains 2 boxes and each box contains 8 pens, you are calculating \((3 \times 2) \times 8\). You can compute this as \((6 \times 8) = 48\) or, using the Associative Property, as \(3 \times (2 \times 8) = 3 \times 16 = 48\). Grouping the numbers differently often makes calculations simpler and faster!

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