Question

1. without computing, select all of the expressions that have the same value as 81·(37 + 59). a. 81·(59 + 37) b. (81·37) + 59 c. (81·37) + (81·59) d. 81 + (37·59) e. (37 + 59)·81

Explanation:

The original expression is \(81 \cdot (37 + 59)\). We can use the distributive property of multiplication over addition, which states that \(a \cdot (b + c)=a\cdot b + a\cdot c\), and also the commutative property of addition (which states that \(b + c=c + b\)) and the commutative property of multiplication (which states that \(a\cdot b = b\cdot a\)) to analyze each option:

Step 1: Analyze Option A

Option A is \(81 \cdot (59 + 37)\). By the commutative property of addition, \(37+59 = 59 + 37\). So, \(81\cdot(37 + 59)=81\cdot(59 + 37)\). Thus, Option A is equivalent.

Step 2: Analyze Option B

Option B is \((81 \cdot 37)+59\). The original expression is \(81\cdot(37 + 59)=81\cdot37+81\cdot59\) (by distributive property). This is not equal to \(81\cdot37 + 59\) (since \(81\cdot59
eq59\) unless \(81 = 1\), which it is not). So, Option B is not equivalent.

Step 3: Analyze Option C

Option C is \((81 \cdot 37)+(81 \cdot 59)\). By the distributive property of multiplication over addition, \(a\cdot(b + c)=a\cdot b+a\cdot c\) (here \(a = 81\), \(b = 37\), \(c = 59\)), so \(81\cdot(37 + 59)=(81\cdot37)+(81\cdot59)\). Thus, Option C is equivalent.

Step 4: Analyze Option D

Option D is \(81+(37 \cdot 59)\). The original expression is a product of 81 and the sum of 37 and 59, while this is a sum of 81 and the product of 37 and 59. These are not equivalent (e.g., \(81\cdot(37 + 59)=81\cdot96 = 7776\) and \(81+(37\cdot59)=81 + 2183=2264\), which are not equal). So, Option D is not equivalent.

Step 5: Analyze Option E

Option E is \((37 + 59)\cdot81\). By the commutative property of multiplication, \(a\cdot b=b\cdot a\) (here \(a = 81\), \(b=(37 + 59)\)), so \(81\cdot(37 + 59)=(37 + 59)\cdot81\). Thus, Option E is equivalent.

Answer:

A. \(81 \cdot (59 + 37)\), C. \((81 \cdot 37)+(81 \cdot 59)\), E. \((37 + 59) \cdot 81\)