To determine which choice is equivalent to the product [tex]\(\sqrt[8]{8} \cdot \sqrt{5}\)[/tex], we will first evaluate this product, then compare it with the given options.
1. Evaluate [tex]\(\sqrt[8]{8}\)[/tex]:
- The eighth root of 8 is:
[tex]\[
\sqrt[8]{8} = 8^{1/8} \approx 1.2968395546510096
\][/tex]
2. Evaluate [tex]\(\sqrt{5}\)[/tex]:
- The square root of 5 is:
[tex]\[
\sqrt{5} \approx 2.23606797749979
\][/tex]
3. Calculate the product [tex]\(\sqrt[8]{8} \cdot \sqrt{5}\)[/tex]:
- Multiplying the above results together:
[tex]\[
\sqrt[8]{8} \cdot \sqrt{5} \approx 1.2968395546510096 \cdot 2.23606797749979 \approx 2.8998214001102114
\][/tex]
4. Compare with the provided options:
- Option A: [tex]\(\sqrt{13}\)[/tex]
[tex]\[
\sqrt{13} \approx 3.605551275463989
\][/tex]
[tex]\[
2.8998214001102114 \neq 3.605551275463989
\][/tex]
This is not equivalent.
- Option B: [tex]\(4 \sqrt{70}\)[/tex]
[tex]\[
4 \sqrt{70} \approx 4 \cdot 8.366600265340756 \approx 33.46640106136302
\][/tex]
[tex]\[
2.8998214001102114 \neq 33.46640106136302
\][/tex]
This is not equivalent.
- Option C: [tex]\(2 \sqrt{10}\)[/tex]
[tex]\[
2 \sqrt{10} \approx 2 \cdot 3.1622776601683795 \approx 6.324555320336759
\][/tex]
[tex]\[
2.8998214001102114 \neq 6.324555320336759
\][/tex]
This is not equivalent.
- Option D: [tex]\(10 \sqrt{2}\)[/tex]
[tex]\[
10 \sqrt{2} \approx 10 \cdot 1.4142135623730951 \approx 14.142135623730951
\][/tex]
[tex]\[
2.8998214001102114 \neq 14.142135623730951
\][/tex]
This is not equivalent.
After comparing the product [tex]\(\sqrt[8]{8} \cdot \sqrt{5}\)[/tex] with each of the provided options, we find that none of the provided options (A, B, C, or D) match the computed product of 2.8998214001102114.
Therefore, none of the given choices (A through D) are equivalent to the product [tex]\(\sqrt[8]{8} \cdot \sqrt{5}\)[/tex].
