To simplify the given expression [tex]\(2 a^3 \sqrt[5]{b^{12} c} + 12 b \sqrt[5]{243 a^{15} b^7 c}\)[/tex], we will evaluate each term separately and then combine and simplify where possible.
### Step 1: Simplify each term individually
1. First Term: [tex]\(2 a^3 \sqrt[5]{b^{12} c}\)[/tex]
Let's rewrite the term in a more manageable form:
[tex]\[
2 a^3 \sqrt[5]{b^{12} c}
\][/tex]
The exponent [tex]\(b^{12} c\)[/tex] under the fifth root can be kept as it is since no further simplification of the individual components is possible.
2. Second Term: [tex]\(12 b \sqrt[5]{243 a^{15} b^7 c}\)[/tex]
Let's first handle the term inside the root:
[tex]\[
243 = 3^5
\][/tex]
Thus, we can write:
[tex]\[
243 a^{15} b^7 c = (3^5) a^{15} b^7 c
\][/tex]
Now, combining under the fifth root:
[tex]\[
\sqrt[5]{243 a^{15} b^7 c} = \sqrt[5]{(3^5) a^{15} b^7 c}
\][/tex]
The fifth root of [tex]\( 3^5 \)[/tex] simplifies to [tex]\( 3 \)[/tex], so we have:
[tex]\[
\sqrt[5]{243 a^{15} b^7 c} = 3 \sqrt[5]{a^{15} b^7 c}
\][/tex]
The entire term becomes:
[tex]\[
12 b \cdot 3 \sqrt[5]{a^{15} b^7 c} = 36 b \sqrt[5]{a^{15} b^7 c}
\][/tex]
### Step 2: Combine the two terms
Now we combine the simplified terms:
[tex]\[
2 a^3 \sqrt[5]{b^{12} c} + 36 b \sqrt[5]{a^{15} b^7 c}
\][/tex]
### Step 3: Simplification of the final expression
Since the terms inside the fifth roots contain distinct components, further simplification by factoring is not straightforward.
Thus, the simplified expression remains:
[tex]\[
2 a^3 \sqrt[5]{b^{12} c} + 36 b \sqrt[5]{a^{15} b^7 c}
\][/tex]
