1. **State the problem:** Find the value of $\left(3x^{2}y^{-5}\right)^{3}$. 2. **Recall the exponent rules:** When raising a product to a power, raise each factor to that power: $$\left(ab\right)^n = a^n b^n$$ Also, when raising a power to another power, multiply the exponents: $$\left(x^m\right)^n = x^{m \times n}$$ 3. **Apply the rules:** $$\left(3x^{2}y^{-5}\right)^3 = 3^3 \times \left(x^{2}\right)^3 \times \left(y^{-5}\right)^3$$ 4. **Calculate each term:** $$3^3 = 27$$ $$\left(x^{2}\right)^3 = x^{2 \times 3} = x^{6}$$ $$\left(y^{-5}\right)^3 = y^{-5 \times 3} = y^{-15}$$ 5. **Combine all:** $$27x^{6}y^{-15}$$ 6. **Final answer:** Option B) $27x^{6}y^{-15}$