To find the value of [tex]\(\sqrt[4]{81 y^{12}}\)[/tex], we need to simplify the given expression step-by-step.
1. Understand the components of the expression:
- [tex]\(81 y^{12}\)[/tex] is the expression inside the fourth root.
- First, we will separate the constants and the variable parts: [tex]\(81\)[/tex] and [tex]\(y^{12}\)[/tex].
2. Deal with the constant term (81):
- We know that [tex]\(81 = 3^4\)[/tex].
- Taking the fourth root of [tex]\(81\)[/tex]:
[tex]\[
\sqrt[4]{81} = \sqrt[4]{3^4} = 3
\][/tex]
3. Deal with the variable term ([tex]\(y^{12}\)[/tex]):
- We can use the property of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
- So, [tex]\(\sqrt[4]{y^{12}}\)[/tex] is equivalent to:
[tex]\[
y^{12/4} = y^3
\][/tex]
4. Combine the results:
- Multiply the results from the constant term and the variable term:
[tex]\[
\sqrt[4]{81 y^{12}} = \sqrt[4]{81} \cdot \sqrt[4]{y^{12}} = 3 \cdot y^3 = 3y^3
\][/tex]
Thus, the final simplified result is:
[tex]\[
3 y^3
\][/tex]
Hence, the correct answer is:
C. [tex]\(3 y^3\)[/tex]
