Tell what property allows you to compute [tex]$\frac{1}{3} \times\left(6 \times \frac{4}{3}\right)$[/tex] as [tex]$\left(\frac{1}{3} \times 6\right) \times \frac{4}{3}$[/tex].
Answer (2)
The property that allows the computation of 3 1 × ( 6 × 3 4 ) as ( 3 1 × 6 ) × 3 4 is the associative property of multiplication, which states that changing the grouping of factors does not change the product. This property expresses that for any numbers a , b , and c , ( (a \times b) \times c = a \times (b \times c). \
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The problem requires identifying the property that allows 3 1 × ( 6 × 3 4 ) to be computed as ( 3 1 × 6 ) × 3 4 .
The associative property of multiplication allows changing the grouping of factors without changing the product.
The associative property is defined as ( a × b ) × c = a × ( b × c ) .
Therefore, the property is the associative property of multiplication. Associative Property of Multiplication
Explanation
Understanding the Problem We are asked to identify the property that allows us to compute 3 1 × ( 6 × 3 4 ) as ( 3 1 × 6 ) × 3 4 .
Identifying the Property The property that allows us to compute 3 1 × ( 6 × 3 4 ) as ( 3 1 × 6 ) × 3 4 is the associative property of multiplication.
Stating the Associative Property The associative property of multiplication states that for any real numbers a , b , and c , the order in which we group the numbers when multiplying does not affect the result. In other words: ( a × b ) × c = a × ( b × c ) In our case, a = 3 1 , b = 6 , and c = 3 4 . So, we have: ( 3 1 × 6 ) × 3 4 = 3 1 × ( 6 × 3 4 )
Final Answer Therefore, the property that allows the given computation is the associative property of multiplication.
Examples
The associative property is useful in many real-life situations. For example, when calculating the total cost of multiple items with discounts, the order in which you apply the discounts and multiply by quantities doesn't change the final amount. This principle ensures that regardless of how you group the calculations, the outcome remains consistent, simplifying complex computations in finance, inventory management, and other fields.
