Which expression is equivalent to [tex]\left(64 y^{100}\right)^{\frac{1}{2}}[/tex]?

A. [tex]8 y^{10}[/tex]
B. [tex]8 y^{50}[/tex]
C. [tex]32 y^{10}[/tex]
D. [tex]32 y^{50}[/tex]



Answer :

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]