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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 $a^m \cdot a^n = a^{m+n}$.

AlgebraExponentsSimplificationProduct of Powers RulePower of a Product Rule
2025/4/24

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=anbn(ab)^n = a^n b^n, and the product of powers rule, which states that am⋅an=am+na^m \cdot a^n = a^{m+n}.

2. Solution Steps

1) (k⋅2k3)3(k \cdot 2k^3)^3
=(2k4)3= (2k^4)^3
=23(k4)3= 2^3 (k^4)^3
=8k4⋅3= 8k^{4 \cdot 3}
=8k12= 8k^{12}
2) (6c5n)4(6c^5n)^4
=64(c5)4n4= 6^4 (c^5)^4 n^4
=1296c5⋅4n4= 1296 c^{5 \cdot 4} n^4
=1296c20n4= 1296 c^{20} n^4
3) (4w2⋅w)3(4w^2 \cdot w)^3
=(4w3)3= (4w^3)^3
=43(w3)3= 4^3 (w^3)^3
=64w3⋅3= 64 w^{3 \cdot 3}
=64w9= 64 w^9
4) (d⋅3d3⋅d3)2(d \cdot 3d^3 \cdot d^3)^2
=(3d1+3+3)2= (3 d^{1+3+3})^2
=(3d7)2= (3d^7)^2
=32(d7)2= 3^2 (d^7)^2
=9d7⋅2= 9d^{7 \cdot 2}
=9d14= 9d^{14}
5) (4dz5)3(4dz^5)^3
=43d3(z5)3= 4^3 d^3 (z^5)^3
=64d3z5⋅3= 64 d^3 z^{5 \cdot 3}
=64d3z15= 64 d^3 z^{15}
6) (2k⋅3k3)2(2k \cdot 3k^3)^2
=(6k4)2= (6k^4)^2
=62(k4)2= 6^2 (k^4)^2
=36k4⋅2= 36 k^{4 \cdot 2}
=36k8= 36 k^8
7) (4n2⋅2n⋅n3)2(4n^2 \cdot 2n \cdot n^3)^2
=(8n2+1+3)2= (8 n^{2+1+3})^2
=(8n6)2= (8 n^6)^2
=82(n6)2= 8^2 (n^6)^2
=64n6⋅2= 64 n^{6 \cdot 2}
=64n12= 64 n^{12}
8) (2h3⋅4h2)2(2h^3 \cdot 4h^2)^2
=(8h3+2)2= (8h^{3+2})^2
=(8h5)2= (8h^5)^2
=82(h5)2= 8^2 (h^5)^2
=64h5⋅2= 64 h^{5 \cdot 2}
=64h10= 64 h^{10}
9) (4g3⋅2g2⋅g)3(4g^3 \cdot 2g^2 \cdot g)^3
=(8g3+2+1)3= (8 g^{3+2+1})^3
=(8g6)3= (8g^6)^3
=83(g6)3= 8^3 (g^6)^3
=512g6⋅3= 512 g^{6 \cdot 3}
=512g18= 512 g^{18}
10) (2n⋅3n2⋅n3)3(2n \cdot 3n^2 \cdot n^3)^3
=(6n1+2+3)3= (6 n^{1+2+3})^3
=(6n6)3= (6 n^6)^3
=63(n6)3= 6^3 (n^6)^3
=216n6⋅3= 216 n^{6 \cdot 3}
=216n18= 216 n^{18}
11) (2y2⋅y3)2(2y^2 \cdot y^3)^2
=(2y2+3)2= (2y^{2+3})^2
=(2y5)2= (2y^5)^2
=22(y5)2= 2^2 (y^5)^2
=4y5⋅2= 4 y^{5 \cdot 2}
=4y10= 4 y^{10}
12) (4b6)4(4b^6)^4
=44(b6)4= 4^4 (b^6)^4
=256b6⋅4= 256 b^{6 \cdot 4}
=256b24= 256 b^{24}
13) (2b3⋅b2⋅3b)3(2b^3 \cdot b^2 \cdot 3b)^3
=(6b3+2+1)3= (6 b^{3+2+1})^3
=(6b6)3= (6b^6)^3
=63(b6)3= 6^3 (b^6)^3
=216b6⋅3= 216 b^{6 \cdot 3}
=216b18= 216 b^{18}
14) (3z3n4)4(3z^3n^4)^4
=34(z3)4(n4)4= 3^4 (z^3)^4 (n^4)^4
=81z3⋅4n4⋅4= 81 z^{3 \cdot 4} n^{4 \cdot 4}
=81z12n16= 81 z^{12} n^{16}

3. Final Answer

1) 8k128k^{12}
2) 1296c20n41296c^{20}n^4
3) 64w964w^9
4) 9d149d^{14}
5) 64d3z1564d^3z^{15}
6) 36k836k^8
7) 64n1264n^{12}
8) 64h1064h^{10}
9) 512g18512g^{18}
10) 216n18216n^{18}
11) 4y104y^{10}
12) 256b24256b^{24}
13) 216b18216b^{18}
14) 81z12n1681z^{12}n^{16}

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