Single Rational Expression D49102

1. **State the problem:** Write the expression $\frac{1}{x^3} \div \frac{2}{x}$ as a single rational expression. 2. **Recall the division rule for fractions:** Dividing by a fraction is the same as multiplying by its reciprocal. So, $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$ 3. **Apply the rule:** $$\frac{1}{x^3} \div \frac{2}{x} = \frac{1}{x^3} \times \frac{x}{2}$$ 4. **Multiply the numerators and denominators:** $$= \frac{1 \times x}{x^3 \times 2} = \frac{x}{2x^3}$$ 5. **Simplify the expression:** Cancel common factors using the property $\frac{x}{x^3} = \frac{\cancel{x}^1}{\cancel{x}^3} = \frac{1}{x^{3-1}} = \frac{1}{x^2}$: $$\frac{x}{2x^3} = \frac{\cancel{x}^1}{2 \times \cancel{x}^3} = \frac{1}{2x^2}$$ 6. **Final answer:** $$\boxed{\frac{1}{2x^2}}$$