The Correct Answer and Explanation is:
The correct answer is determined by simplifying each of the four mathematical expressions and then comparing the results.
Different answer: B
Same answer: A and D
Here is a detailed explanation of how to arrive at these answers:
The problem asks us to evaluate four different expressions and identify which one yields a different result and which ones yield the same result. Let’s analyze each option step by step.
1. Simplify 3³ ⋅ 3⁶
This expression involves the multiplication of two powers with the same base, which is 3. According to the product of powers rule in exponentiation (xᵃ ⋅ xᵇ = xᵃ⁺ᵇ), we add the exponents.
3³ ⋅ 3⁶ = 3⁽³⁺⁶⁾ = 3⁹.
2. Simplify 3³ + 6
This expression involves an exponent and an addition. Following the order of operations, we must calculate the exponent first.
3³ = 3 × 3 × 3 = 27.
Next, we perform the addition:
27 + 6 = 33.
3. Simplify 3⁶ ⋅ 3
This is another multiplication of powers with the same base. The term ‘3’ can be written as 3¹. Applying the same product of powers rule:
3⁶ ⋅ 3¹ = 3⁽⁶⁺¹⁾ = 3⁷.
4. Simplify 3⁶ ⋅ 3³
Similar to the first option, we apply the product of powers rule by adding the exponents.
3⁶ ⋅ 3³ = 3⁽⁶⁺³⁾ = 3⁹.
Comparison of the Results
Now, let’s compare the simplified form of each expression:
- Result A: 3⁹
- Result B: 33
- Result C: 3⁷
- Result D: 3⁹
From this comparison, we can draw two conclusions.
First, the expressions in options A and D produce the same result: 3⁹. This is because multiplication is commutative (a ⋅ b = b ⋅ a), so 3³ ⋅ 3⁶ is equivalent to 3⁶ ⋅ 3³. Therefore, A and D have the same answer.
Second, option B is the one that is fundamentally different. While options A, C, and D are all multiplications that simplify to a power of 3, option B is an addition problem. Its result, 33, is not a simple power of 3. While option C also has a unique numerical value (3⁷ = 2,187), option B is the most distinct in terms of the mathematical operation used and the form of its result. Thus, B is the “different answer”.
