1. **State the problem:** Simplify the expression $$\frac{15bc^2 \times 2b}{6b^2}$$. 2. **Write the expression:** $$\frac{15bc^2 \times 2b}{6b^2}$$. 3. **Multiply the numerator:** $$15bc^2 \times 2b = 30b^2c^2$$. 4. **Rewrite the expression:** $$\frac{30b^2c^2}{6b^2}$$. 5. **Simplify the fraction by canceling common factors:** $$\frac{\cancel{30}b^2c^2}{\cancel{6}b^2} = \frac{5\cancel{b^2}c^2}{\cancel{b^2}} = 5c^2$$. 6. **Final answer:** $$5c^2$$. --- 1. **State the problem:** Show that $$(2y^3)^3 \equiv 8y^9$$. 2. **Use the power of a product rule:** $$(ab)^n = a^n b^n$$. 3. **Apply the rule:** $$(2y^3)^3 = 2^3 (y^3)^3$$. 4. **Calculate powers:** $$2^3 = 8$$ and $$(y^3)^3 = y^{3 \times 3} = y^9$$. 5. **Combine results:** $$8y^9$$. 6. **Final answer:** $$(2y^3)^3 \equiv 8y^9$$.