12. \( \frac{4 y}{\sqrt{2 x^{4} y}} \)

To simplify the expression \( \frac{4 y}{\sqrt{2 x^{4} y}} \), you first need to deal with the square root in the denominator. The square root can be separated as follows: \[ \sqrt{2 x^{4} y} = \sqrt{2} \cdot \sqrt{x^{4}} \cdot \sqrt{y} = \sqrt{2} \cdot x^{2} \cdot \sqrt{y} \] Now substitute this back into the original expression: \[ \frac{4 y}{\sqrt{2} \cdot x^{2} \cdot \sqrt{y}} \] Next, simplify by separating \(y\): \[ \frac{4 \cancel{y}}{\sqrt{2} \cdot x^{2} \cdot \sqrt{y}} = \frac{4 \sqrt{y}}{\sqrt{2} \cdot x^{2}} \] And that's the simplified version of the expression: \[ \frac{4 \sqrt{y}}{\sqrt{2} \cdot x^{2}} \] Fun fact! The process of simplifying radical expressions has roots in ancient civilizations, where they explored square roots and arithmetic operations long before formal algebra was developed. These early mathematicians laid the groundwork for much of the algebra we use today, giving rise to methods for simplifying and manipulating expressions that are still taught in classrooms around the world. When it comes to handling radicals, a common mistake is forgetting to simplify the square root fully. It’s essential to break down terms completely and to be cautious about how you manipulate \( \sqrt{a \cdot b} \) into \( \sqrt{a} \cdot \sqrt{b} \). Always check for simplification opportunities in both the numerator and denominator before finalizing your answer! Happy simplifying!