To simplify the expression [tex]\(\left(3 a^4 b^4 c^5\right)^2\left(2 a^4 b^4 c^3\right)^3\)[/tex] completely, we will follow these steps:
### Step 1: Expand Each Part
First, we expand each part of the expression individually.
#### Part 1: Expand [tex]\(\left(3 a^4 b^4 c^5\right)^2\)[/tex]
Using the power rule [tex]\((xy)^n = x^n y^n\)[/tex]:
[tex]\[
\left(3 a^4 b^4 c^5\right)^2 = 3^2 \cdot (a^4)^2 \cdot (b^4)^2 \cdot (c^5)^2
\][/tex]
Calculate each component:
[tex]\[
3^2 = 9
\][/tex]
[tex]\[
(a^4)^2 = a^{4 \cdot 2} = a^8
\][/tex]
[tex]\[
(b^4)^2 = b^{4 \cdot 2} = b^8
\][/tex]
[tex]\[
(c^5)^2 = c^{5 \cdot 2} = c^{10}
\][/tex]
Thus, the expanded form is:
[tex]\[
\left(3 a^4 b^4 c^5\right)^2 = 9 a^8 b^8 c^{10}
\][/tex]
#### Part 2: Expand [tex]\(\left(2 a^4 b^4 c^3\right)^3\)[/tex]
Using the power rule again:
[tex]\[
\left(2 a^4 b^4 c^3\right)^3 = 2^3 \cdot (a^4)^3 \cdot (b^4)^3 \cdot (c^3)^3
\][/tex]
Calculate each component:
[tex]\[
2^3 = 8
\][/tex]
[tex]\[
(a^4)^3 = a^{4 \cdot 3} = a^{12}
\][/tex]
[tex]\[
(b^4)^3 = b^{4 \cdot 3} = b^{12}
\][/tex]
[tex]\[
(c^3)^3 = c^{3 \cdot 3} = c^9
\][/tex]
Thus, the expanded form is:
[tex]\[
\left(2 a^4 b^4 c^3\right)^3 = 8 a^{12} b^{12} c^9
\][/tex]
### Step 2: Multiply the Expanded Parts Together
Now, multiply the expanded parts together:
[tex]\[
9 a^8 b^8 c^{10} \cdot 8 a^{12} b^{12} c^9
\][/tex]
First, multiply the numerical coefficients:
[tex]\[
9 \cdot 8 = 72
\][/tex]
Next, multiply the variables with the same base using the rule [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
a^8 \cdot a^{12} = a^{8+12} = a^{20}
\][/tex]
[tex]\[
b^8 \cdot b^{12} = b^{8+12} = b^{20}
\][/tex]
[tex]\[
c^{10} \cdot c^9 = c^{10+9} = c^{19}
\][/tex]
### Final Simplified Expression
Combining all these results together, we get the completely simplified expression:
[tex]\[
72 a^{20} b^{20} c^{19}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
\boxed{72 a^{20} b^{20} c^{19}}
\][/tex]
