The problem asks to simplify expressions involving exponents and products. We need to apply the power of a product rule, which states that $(ab)^n = a^n b^n$, and the product of powers rule, which states that $a^m \cdot a^n = a^{m+n}$.

1. Problem Description

The problem asks to simplify expressions involving exponents and products. We need to apply the power of a product rule, which states that $(ab)^n = a^n b^n$, and the product of powers rule, which states that $a^m \cdot a^n = a^{m+n}$.

2. Solution Steps

1) $(k \cdot 2k^3)^3$

$= (2k^4)^3$

$= 2^3 (k^4)^3$

$= 8k^{4 \cdot 3}$

$= 8k^{12}$

2) $(6c^5n)^4$

$= 6^4 (c^5)^4 n^4$

$= 1296 c^{5 \cdot 4} n^4$

$= 1296 c^{20} n^4$

3) $(4w^2 \cdot w)^3$

$= (4w^3)^3$

$= 4^3 (w^3)^3$

$= 64 w^{3 \cdot 3}$

$= 64 w^9$

4) $(d \cdot 3d^3 \cdot d^3)^2$

$= (3 d^{1+3+3})^2$

$= (3d^7)^2$

$= 3^2 (d^7)^2$

$= 9d^{7 \cdot 2}$

$= 9d^{14}$

5) $(4dz^5)^3$

$= 4^3 d^3 (z^5)^3$

$= 64 d^3 z^{5 \cdot 3}$

$= 64 d^3 z^{15}$

6) $(2k \cdot 3k^3)^2$

$= (6k^4)^2$

$= 6^2 (k^4)^2$

$= 36 k^{4 \cdot 2}$

$= 36 k^8$

7) $(4n^2 \cdot 2n \cdot n^3)^2$

$= (8 n^{2+1+3})^2$

$= (8 n^6)^2$

$= 8^2 (n^6)^2$

$= 64 n^{6 \cdot 2}$

$= 64 n^{12}$

8) $(2h^3 \cdot 4h^2)^2$

$= (8h^{3+2})^2$

$= (8h^5)^2$

$= 8^2 (h^5)^2$

$= 64 h^{5 \cdot 2}$

$= 64 h^{10}$

9) $(4g^3 \cdot 2g^2 \cdot g)^3$

$= (8 g^{3+2+1})^3$

$= (8g^6)^3$

$= 8^3 (g^6)^3$

$= 512 g^{6 \cdot 3}$

$= 512 g^{18}$

10) $(2n \cdot 3n^2 \cdot n^3)^3$

$= (6 n^{1+2+3})^3$

$= (6 n^6)^3$

$= 6^3 (n^6)^3$

$= 216 n^{6 \cdot 3}$

$= 216 n^{18}$

11) $(2y^2 \cdot y^3)^2$

$= (2y^{2+3})^2$

$= (2y^5)^2$

$= 2^2 (y^5)^2$

$= 4 y^{5 \cdot 2}$

$= 4 y^{10}$

12) $(4b^6)^4$

$= 4^4 (b^6)^4$

$= 256 b^{6 \cdot 4}$

$= 256 b^{24}$

13) $(2b^3 \cdot b^2 \cdot 3b)^3$

$= (6 b^{3+2+1})^3$

$= (6b^6)^3$

$= 6^3 (b^6)^3$

$= 216 b^{6 \cdot 3}$

$= 216 b^{18}$

14) $(3z^3n^4)^4$

$= 3^4 (z^3)^4 (n^4)^4$

$= 81 z^{3 \cdot 4} n^{4 \cdot 4}$

$= 81 z^{12} n^{16}$

3. Final Answer

1) $8k^{12}$

2) $1296c^{20}n^4$

3) $64w^9$

4) $9d^{14}$

5) $64d^3z^{15}$

6) $36k^8$

7) $64n^{12}$

8) $64h^{10}$

9) $512g^{18}$

10) $216n^{18}$

11) $4y^{10}$

12) $256b^{24}$

13) $216b^{18}$

14) $81z^{12}n^{16}$