Completely simplify the rational expression $\frac{18 x^3 y}{6 x y^3}$. State any restrictions on the variables.
Answer (2)
The simplified form of the rational expression 6 x y 3 18 x 3 y is y 2 3 x 2 , with the restrictions that x = 0 and y = 0 .
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Simplify the numerical coefficients: 6 18 = 3 .
Simplify the x terms: x x 3 = x 3 − 1 = x 2 .
Simplify the y terms: y 3 y = y 1 − 3 = y − 2 = y 2 1 .
Combine the simplified terms and state the restrictions: y 2 3 x 2 , x = 0 , y = 0 .
y 2 3 x 2 ; x = 0 , y = 0
Explanation
Understanding the Problem We are asked to simplify the rational expression 6 x y 3 18 x 3 y and state any restrictions on the variables. This involves simplifying the numerical coefficients and the variable terms separately, and then combining the results. We also need to identify any values of the variables that would make the denominator zero, as these values are not allowed.
Simplifying Coefficients First, let's simplify the numerical coefficients. We have 6 18 = 3 .
Simplifying x Terms Next, let's simplify the x terms. We have x x 3 = x 3 − 1 = x 2 .
Simplifying y Terms Now, let's simplify the y terms. We have y 3 y = y 1 − 3 = y − 2 = y 2 1 .
Combining Terms Combining the simplified terms, we get 3 ⋅ x 2 ⋅ y 2 1 = y 2 3 x 2 .
Identifying Restrictions Finally, let's identify any restrictions on the variables. Since x and y appear in the denominator of the original expression, they cannot be equal to zero. Therefore, x = 0 and y = 0 .
Final Answer Therefore, the simplified rational expression is y 2 3 x 2 , and the variable restrictions are x = 0 , y = 0 .
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 allows engineers to analyze and design circuits more efficiently. Similarly, in physics, rational expressions can be used to describe the motion of objects or the behavior of waves. Understanding how to simplify these expressions is crucial for solving problems and making predictions in these fields. For instance, if we consider the formula for combined resistance R = R 1 1 + R 2 1 1 , we can simplify it to R = R 1 + R 2 R 1 R 2 .
