Which expression is equivalent to the given expression?
$\frac{\left(3 m^2 n\right)^3}{m n^4}$
A. $\frac{9 m^5}{n}$
B. $27 m^4 n$
C. $\frac{27 m^5}{n}$
D. $9 m^4 n$
Answer (1)
Apply the power of a product rule to the numerator: ( 3 m 2 n ) 3 = 3 3 ( m 2 ) 3 n 3 = 27 m 6 n 3 .
Divide the numerator by the denominator: m n 4 27 m 6 n 3 = 27 ⋅ m m 6 ⋅ n 4 n 3 .
Simplify the expression by subtracting exponents: 27 m 6 − 1 n 3 − 4 = 27 m 5 n − 1 .
Rewrite the expression with positive exponents: 27 m 5 n − 1 = n 27 m 5 .
The equivalent expression is n 27 m 5 .
Explanation
Understanding the Problem We are asked to find an expression equivalent to m n 4 ( 3 m 2 n ) 3 . Let's simplify the given expression step by step.
Applying the Power of a Product Rule First, we apply the power of a product rule to the numerator: ( ab ) n = a n b n . This gives us m n 4 ( 3 m 2 n ) 3 = m n 4 3 3 ( m 2 ) 3 n 3
Simplifying the Numerator Next, we simplify 3 3 and apply the power of a power rule ( a m ) n = a mn to ( m 2 ) 3 . Since 3 3 = 27 and ( m 2 ) 3 = m 2 × 3 = m 6 , we have m n 4 3 3 ( m 2 ) 3 n 3 = m n 4 27 m 6 n 3
Dividing and Subtracting Exponents Now, we divide the numerator by the denominator. When dividing terms with the same base, we subtract the exponents: a n a m = a m − n . So, m m 6 = m 6 − 1 = m 5 and n 4 n 3 = n 3 − 4 = n − 1 . Therefore, m n 4 27 m 6 n 3 = 27 m 6 − 1 n 3 − 4 = 27 m 5 n − 1
Simplifying the Expression Finally, we rewrite n − 1 as n 1 , which gives us 27 m 5 n − 1 = 27 m 5 ⋅ n 1 = n 27 m 5 Thus, the expression equivalent to m n 4 ( 3 m 2 n ) 3 is n 27 m 5 .
Final Answer The equivalent expression is n 27 m 5 . Therefore, the correct answer is C.
Examples
Understanding how to simplify expressions with exponents is crucial in many areas of science and engineering. For example, when calculating the volume of a complex 3D shape or determining the electrical resistance in a circuit, you often need to manipulate expressions with exponents. Simplifying these expressions correctly ensures accurate results and efficient problem-solving in real-world applications.
