Multiply the following rational expressions. Write the answer in \[ \frac{15 x^{2}}{7} \cdot \frac{35}{6 x} \] \( \frac{15 x^{2}}{7} \cdot \frac{35}{6 x}=\square \) (Simplify your answer. Use integers or fractions for any number:

To multiply the rational expressions, start by multiplying the numerators and the denominators: \[ \frac{15 x^{2}}{7} \cdot \frac{35}{6 x} = \frac{15 \cdot 35 x^{2}}{7 \cdot 6 x} \] Calculating the numerators gives: \[ 15 \cdot 35 = 525 \] And for the denominators: \[ 7 \cdot 6 = 42 \] Thus, we have: \[ \frac{525 x^{2}}{42 x} \] Next, simplify by dividing both the numerator and denominator by 21 (the greatest common divisor): \[ 525 \div 21 = 25, \quad 42 \div 21 = 2 \] This simplifies our expression to: \[ \frac{25 x}{2} \] So, the final answer is: \[ \frac{25 x}{2} = \square \] Now onto some fun facts! Rational expressions can often be simplified by finding common factors in the numerator and denominator. This is not only useful for simplifying fractions but also aids in understanding how to manipulate algebraic expressions more effectively. When it comes to real-world applications, think of rational expressions as recipes! Just like you scale ingredients up or down, you can use multiplication of rational expressions to adjust portions in cooking or even when mixing solutions in chemistry. It's all about balance!