Final answer:
To simplify the expression (3m²n³)³, we cube the number 3 to get 27, and multiply each variable's exponent by 3, resulting in 27m¶n¹. The correct simplification, assuming the correct raising of exponents, is not an option given in the question. C is correct answer .
Step-by-step explanation:
To simplify the expression (3m²n³)³, we need to cube each part of the expression inside the parentheses. This means we will cube the coefficient (3), and multiply the exponent of each variable by 3. The rules for cubing of exponentials indicate that the digit term is cubed normally, and the exponent of the exponential term is multiplied by the power. We know that raising a number to a power is a mathematical shorthand for multiplication of that number by itself as many times as the power indicates.
First, let's cube the numerical coefficient: 3³ = 27.
Next, we apply the exponent rule for the variables: to cube m², we multiply the exponent by 3 (m²³ = m¶), and to cube n³, we multiply that exponent by 3 as well (n³³ = n¹). Thus, m²³ = m¶ and n³³ = n¹.
Finally, we combine these results to complete the simplification of the expression:
(3m²n³)³ = 3³ · m²³ · n³³ = 27m¶n¹, which is none of the options provided in the question. However, there seems to be a typo in the exponent for n. It should be n¹³ (27m¶n¹³).
Based on the given options, the closest correct option would be D. 9m²n¹², which seems to be another typo because 9 should be 27 and m² should be m¶.