To determine which expression is equivalent to [tex]\(10 \sqrt{5}\)[/tex], we will evaluate each given option:
Option A: [tex]\(\sqrt{500}\)[/tex]
First, let's simplify [tex]\(\sqrt{500}\)[/tex]:
[tex]\[ \sqrt{500} = \sqrt{100 \times 5} \][/tex]
We know that [tex]\(\sqrt{100}\)[/tex] is [tex]\(10\)[/tex], so:
[tex]\[ \sqrt{500} = \sqrt{100 \times 5} = \sqrt{100} \times \sqrt{5} = 10 \sqrt{5} \][/tex]
Therefore, [tex]\(\sqrt{500}\)[/tex] simplifies to [tex]\(10 \sqrt{5}\)[/tex], and thus it is equivalent to [tex]\(10 \sqrt{5}\)[/tex].
Option B: [tex]\(\sqrt{105}\)[/tex]
Next, we examine [tex]\(\sqrt{105}\)[/tex]. There are no perfect squares that are factors of 105:
[tex]\[ 105 = 3 \times 5 \times 7 \][/tex]
Since none of these factors are perfect squares, [tex]\(\sqrt{105}\)[/tex] cannot be simplified to [tex]\(10 \sqrt{5}\)[/tex].
Option C: [tex]\(\sqrt{50}\)[/tex]
Now, let's simplify [tex]\(\sqrt{50}\)[/tex]:
[tex]\[ \sqrt{50} = \sqrt{25 \times 2} \][/tex]
We know that [tex]\(\sqrt{25}\)[/tex] is [tex]\(5\)[/tex], so:
[tex]\[ \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5 \sqrt{2} \][/tex]
Comparing [tex]\(5 \sqrt{2}\)[/tex] to [tex]\(10 \sqrt{5}\)[/tex], they are clearly not equivalent.
Option D: [tex]\(\sqrt{15}\)[/tex]
Finally, let's examine [tex]\(\sqrt{15}\)[/tex]. There are no perfect squares that are factors of 15:
[tex]\[ 15 = 3 \times 5 \][/tex]
Since none of these factors are perfect squares, [tex]\(\sqrt{15}\)[/tex] cannot be simplified to [tex]\(10 \sqrt{5}\)[/tex].
Conclusion:
The only option that is equivalent to [tex]\(10 \sqrt{5}\)[/tex] is:
A. [tex]\(\sqrt{500}\)[/tex]
