Perform the indicated operation of multiplication or division on the rational expressions and simplify.
$\frac{9 x^2}{6 y} \cdot \frac{12 y^2}{3 x^2}$
Answer (2)
Multiply the numerators and denominators: 6 y 9 x 2 ⋅ 3 x 2 12 y 2 = 18 x 2 y 108 x 2 y 2 .
Cancel the x 2 terms: 18 x 2 y 108 x 2 y 2 = 18 y 108 y 2 .
Cancel the y terms: 18 y 108 y 2 = 18 108 y .
Simplify the numerical coefficients: 18 108 y = 6 y . The final answer is 6 y .
Explanation
Understanding the problem We are given the expression 6 y 9 x 2 ⋅ 3 x 2 12 y 2 and we need to simplify it.
Multiplying the expressions First, multiply the numerators and the denominators: 6 y 9 x 2 ⋅ 3 x 2 12 y 2 = 6 y ⋅ 3 x 2 9 x 2 ⋅ 12 y 2 = 18 x 2 y 108 x 2 y 2 .
Cancelling x 2 Now, we simplify the expression by canceling out common factors. We can cancel out x 2 from the numerator and denominator: 18 x 2 y 108 x 2 y 2 = 18 y 108 y 2 .
Cancelling y Next, we cancel out y from the numerator and denominator: 18 y 108 y 2 = 18 108 y .
Simplifying the coefficients Finally, we simplify the numerical coefficients by dividing 108 by 18: 18 108 y = 6 y .
Final Answer Therefore, the simplified expression is 6 y .
Examples
Rational expressions are useful in many real-world applications, such as calculating rates, proportions, and scaling recipes. For example, if you are increasing the size of a cake recipe, you can use rational expressions to determine the amount of each ingredient needed. If the original recipe calls for 2 1 cup of flour and you want to triple the recipe, you would multiply 2 1 by 3 to get 2 3 or 1 2 1 cups of flour.
To simplify the expression 6 y 9 x 2 ⋅ 3 x 2 12 y 2 , we multiply the numerators and denominators, cancel common factors, and simplify coefficients to find the final result of 6 y .
;
