Question:
If $*$ is the operation defined by $a * b = a^b$ for $a, b \in \mathbb{N}$, then $(2 * 3) * 2$ is equal to:
If $*$ is the operation defined by $a * b = a^b$ for $a, b \in \mathbb{N}$, then $(2 * 3) * 2$ is equal to:
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Note that this specific operation (exponentiation) is not associative. This means $(a * b) * c \neq a * (b * c)$. In this problem, $(2 * 3) * 2 = 64$, but $2 * (3 * 2) = 2^9 = 512$. Always follow the order of operations indicated by the parentheses.
Updated On: May 2, 2026
- $81$
- $512$
- $216$
- $64$
- $243$
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The Correct Option is D
Solution and Explanation
Concept:
A binary operation $*$ on a set is a rule that assigns to each pair of elements in the set another element of that same set. In this case, the operation is defined as exponentiation, where the first number is the base and the second number is the exponent.
Step 1: Evaluate the operation inside the parentheses first.
According to the definition $a * b = a^b$: \[ 2 * 3 = 2^3 \] \[ 2 * 3 = 8 \]
Step 2: Perform the operation with the remaining element.
Now substitute the result back into the original expression $(2 * 3) * 2$: \[ 8 * 2 = 8^2 \] \[ 8 * 2 = 64 \]
Step 1: Evaluate the operation inside the parentheses first.
According to the definition $a * b = a^b$: \[ 2 * 3 = 2^3 \] \[ 2 * 3 = 8 \]
Step 2: Perform the operation with the remaining element.
Now substitute the result back into the original expression $(2 * 3) * 2$: \[ 8 * 2 = 8^2 \] \[ 8 * 2 = 64 \]
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