To determine which expression is equivalent to [tex]\(\left(64 y^{100}\right)^{\frac{1}{2}}\)[/tex], we can follow these steps:
1. Understand the expression: We have the expression [tex]\(\left(64 y^{100}\right)^{\frac{1}{2}}\)[/tex].
2. Apply the exponent outside the parenthesis: We need to distribute the [tex]\(\frac{1}{2}\)[/tex] exponent to both the numerical coefficient (64) and the variable part ([tex]\(y^{100}\)[/tex]) separately.
3. Simplifying the numerical part:
- Calculate [tex]\((64)^{\frac{1}{2}}\)[/tex].
- The square root of 64 ([tex]\((64)^{\frac{1}{2}}\)[/tex]) is 8.
4. Simplifying the variable part:
- Calculate [tex]\((y^{100})^{\frac{1}{2}}\)[/tex].
- Use the power rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[(y^{100})^{\frac{1}{2}} = y^{100 \cdot \frac{1}{2}} = y^{50}.\][/tex]
5. Combine the simplified parts:
- Multiply the results together: [tex]\(8 \cdot y^{50}\)[/tex].
So, the equivalent expression is [tex]\(8 y^{50}\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{8 y^{50}}. \][/tex]
