Given the equation \( \frac{m^3}{n^6} = \frac{1}{27} \), compare the following quantities:
Quantity A: \(3m\)
Quantity B: \(n^2\)
A. Quantity A is greater.
B. Quantity B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.
Answer (2)
After manipulating the given equation, we find that 3 m is equal to n 2 . Therefore, the two quantities are equal. The correct answer is C.
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To compare Quantity A: 3 m and Quantity B: n 2 , we start from the given equation:
n 6 m 3 = 27 1
Step 1: Rewrite the equation in terms of powers of 3. We know that 27 = 3 3 , so:
n 6 m 3 = 3 − 3
Step 2: Equate the exponents for equality:
Since both sides are in terms of powers of 3, we equate the exponents:
m 3 = 3 − 3 × n 6
m 3 = n 6 × 3 − 3
Step 3: Simplify to find m in terms of n .
Taking the cube root on both sides:
m = ( n 6 × 3 − 3 ) 1/3
m = n 2 × 3 − 1
Step 4: Substitute to compare 3 m and n 2 .
Using the relationship derived, multiply m by 3:
3 m = 3 ( n 2 × 3 − 1 )
3 m = n 2
Therefore, Quantity A, 3 m , is equal to Quantity B, n 2 .
Thus, the correct choice is:
C. The two quantities are equal.
