Perform the indicated operation of multiplication or division on the rational expressions and simplify.
[tex]\frac{21 m}{2 n^3} \div \frac{7 m}{4 n^2}[/tex]
Answer (2)
To simplify the expression 2 n 3 21 m ÷ 4 n 2 7 m , we convert the division to multiplication by the reciprocal, multiply the numerators and denominators, and then simplify to get n 6 .
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Rewrite the division as multiplication by the reciprocal: 2 n 3 21 m × 7 m 4 n 2 .
Multiply the numerators and denominators: 2 n 3 × 7 m 21 m × 4 n 2 = 14 m n 3 84 m n 2 .
Simplify the numerical coefficients: 14 84 = 6 .
Cancel the common factors of m and n : m m = 1 and n 3 n 2 = n 1 , resulting in the final simplified expression: n 6 .
Explanation
Understanding the Problem We are asked to divide two rational expressions and simplify the result. The given expression is 2 n 3 21 m ÷ 4 n 2 7 m .
Rewriting as Multiplication To divide rational expressions, we multiply by the reciprocal of the second expression. So, we rewrite the division as multiplication: 2 n 3 21 m ÷ 4 n 2 7 m = 2 n 3 21 m × 7 m 4 n 2
Multiplying Numerators and Denominators Now, we multiply the numerators and the denominators: 2 n 3 × 7 m 21 m × 4 n 2 = 14 m n 3 84 m n 2
Simplifying the Expression Next, we simplify the expression by canceling common factors. First, we simplify the numerical coefficients: 14 84 = 6 Then, we cancel the common factors of m and n : m m = 1 n 3 n 2 = n 1
Final Simplification Combining these simplifications, we get: 14 m n 3 84 m n 2 = 6 × 1 × n 1 = n 6
Final Answer Therefore, the simplified expression is n 6 .
Examples
Rational expressions are used in various fields, such as physics and engineering, to model relationships between different quantities. For example, in electrical engineering, the impedance of a circuit can be represented as a rational expression involving the frequency of the signal. Simplifying these expressions helps in analyzing the behavior of the circuit and designing it efficiently. Similarly, in physics, rational expressions can be used to describe the motion of objects or the behavior of waves. Understanding how to manipulate and simplify these expressions is crucial for solving problems in these fields.
