To determine which expression is equivalent to [tex]\(10 \sqrt{5}\)[/tex], we will evaluate each of the given options and compare the outcomes.
1. Evaluate [tex]\(\sqrt{500}\)[/tex]:
The square root of 500 is approximately:
[tex]\[
\sqrt{500} \approx 22.360679774997898
\][/tex]
2. Evaluate [tex]\(\sqrt{105}\)[/tex]:
The square root of 105 is approximately:
[tex]\[
\sqrt{105} \approx 10.246950765959598
\][/tex]
3. Evaluate [tex]\(\sqrt{50}\)[/tex]:
The square root of 50 is approximately:
[tex]\[
\sqrt{50} \approx 7.0710678118654755
\][/tex]
4. Evaluate [tex]\(\sqrt{15}\)[/tex]:
The square root of 15 is approximately:
[tex]\[
\sqrt{15} \approx 3.872983346207417
\][/tex]
Now, compute [tex]\(10 \sqrt{5}\)[/tex]:
[tex]\[
10 \sqrt{5} \approx 10 \times 2.236067977499789 \approx 22.360679774997898
\][/tex]
Comparing this result to the evaluated expressions above, we see that:
[tex]\[
10 \sqrt{5} \approx 22.360679774997898
\][/tex]
is equal to [tex]\(\sqrt{500}\)[/tex].
Hence, the correct answer is:
[tex]\[
\boxed{\sqrt{500}}
\][/tex]
