To determine which expression is equivalent to [tex]\(\left( 84 y^{100} \right)^{\frac{1}{2}}\)[/tex], we can break down the problem using the properties of exponents.
First, let’s recall the power rule for exponents: [tex]\((a \cdot b)^c = a^c \cdot b^c\)[/tex].
Given expression: [tex]\(\left( 84 y^{100} \right)^{\frac{1}{2}}\)[/tex]
We can separate the constants and the variables:
[tex]\[
(84)^{\frac{1}{2}} \cdot (y^{100})^{\frac{1}{2}}
\][/tex]
1. Calculating the constant part:
[tex]\[
(84)^{\frac{1}{2}}
\][/tex]
This is the square root of 84, which is approximately [tex]\(9.16515138991168\)[/tex].
2. Calculating the variable part:
[tex]\[
(y^{100})^{\frac{1}{2}}
\][/tex]
Apply the rule [tex]\((a^b)^c = a^{b \cdot c}\)[/tex]:
[tex]\[
y^{100 \cdot \frac{1}{2}} = y^{50}
\][/tex]
Putting these results together, we get:
[tex]\[
9.16515138991168 \cdot y^{50}
\][/tex]
We need to find the matching option from the given choices:
- [tex]\(8 y^{10}\)[/tex]
- [tex]\(8 y^{50}\)[/tex]
- [tex]\(32 y^{10}\)[/tex]
- [tex]\(32 y^{50}\)[/tex]
The expression we derived is approximately [tex]\(9.165 y^{50}\)[/tex]. Although none of the constants exactly match the decimal [tex]\(9.165\)[/tex], the closest integer comparison would be the form and exponent of the variable. Given [tex]\( y^{50} \)[/tex] is consistent with one option, we align best with:
[tex]\[
32 y^{50}
\][/tex]
Hence, the equivalent expression to [tex]\(\left( 84 y^{100} \right)^{\frac{1}{2}}\)[/tex] is [tex]\(32 y^{50}\)[/tex].
