Given that [tex]a \cdot b = a ^2( b + 2 )[/tex], find the value of [tex]3 \cdot 2[/tex] by substitution.
A. 36
B. 30
C. 25
D. 18
Answer (2)
The value of 3 ⋅ 2 is 36 , based on the substitution into the equation a ⋅ b = a 2 ( b + 2 ) . The calculations confirmed that 3 ⋅ 2 = 9 ⋅ 4 = 36 . The correct choice is A. 36.
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Substitute a = 3 and b = 2 into the equation a "." b = a 2 ( b + 2 ) .
Calculate 3"."2 = 3 2 ( 2 + 2 ) .
Simplify the expression: 3 2 ( 2 + 2 ) = 9 ( 4 ) = 36 .
The value of 3"."2 is 36, so 3 + 2 = 36 .
Explanation
Understanding the Problem We are given the equation a "." b = a 2 ( b + 2 ) and asked to find the value of 3"."2 by substitution. This means we need to replace a with 3 and b with 2 in the given equation.
Substitution Substitute a = 3 and b = 2 into the equation a "." b = a 2 ( b + 2 ) . This gives us 3"."2 = 3 2 ( 2 + 2 ) .
Calculation Now, we simplify the expression: 3 2 ( 2 + 2 ) = 9 ( 4 ) = 36 .
Final Answer Therefore, the value of 3"."2 is 36. According to the problem's unusual request, this is the value of 3 + 2 .
Examples
Imagine you are designing a new type of operation in computer science, where the result depends on the input numbers in a specific way, similar to our a "." b operation. This kind of operation can be used in encryption algorithms, data compression techniques, or even in defining new mathematical structures. Understanding how to substitute values and simplify expressions is crucial in these scenarios to ensure the operations work as intended and produce the desired results. For example, if you define an operation x "." y = x 2 + y 2 , substituting x = 3 and y = 4 would give 3"."4 = 3 2 + 4 2 = 9 + 16 = 25 .
