1. **State the problem:** Simplify the expression $$\frac{12x^{61}}{7y^{15}} \cdot \frac{y}{8x^{27}}$$. 2. **Write the formula and rules:** When multiplying fractions, multiply numerators together and denominators together: $$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$ Also, use the laws of exponents: $$x^m \cdot x^n = x^{m+n}$$ and $$\frac{x^m}{x^n} = x^{m-n}$$. 3. **Multiply numerators and denominators:** $$\frac{12x^{61}}{7y^{15}} \cdot \frac{y}{8x^{27}} = \frac{12x^{61} \cdot y}{7y^{15} \cdot 8x^{27}} = \frac{12xyx^{61}}{56y^{15}x^{27}}$$ 4. **Simplify coefficients:** $$\frac{12}{56} = \frac{\cancel{12}^{3}}{\cancel{56}^{14}} = \frac{3}{14}$$ 5. **Simplify variables:** - For $x$: $$\frac{x^{61}}{x^{27}} = x^{61-27} = x^{34}$$ - For $y$: $$\frac{y}{y^{15}} = y^{1-15} = y^{-14} = \frac{1}{y^{14}}$$ 6. **Combine all:** $$\frac{3}{14} \cdot x^{34} \cdot \frac{1}{y^{14}} = \frac{3x^{34}}{14y^{14}}$$ **Final answer:** $$\boxed{\frac{3x^{34}}{14y^{14}}}$$