Question

select the correct answer.
which expression is equivalent to $10\sqrt{5}$?
\\(\bigcirc\\) a. \\(\sqrt{500}\\)
\\(\bigcirc\\) b. \\(\sqrt{105}\\)
\\(\bigcirc\\) c. \\(\sqrt{50}\\)
\\(\bigcirc\\) d. \\(\sqrt{15}\\)

Explanation:

Step1: Recall the property of square roots

We know that \(a\sqrt{b}=\sqrt{a^{2}\times b}\) for \(a\geq0\) and \(b\geq0\).

Step2: Apply the property to \(10\sqrt{5}\)

Here, \(a = 10\) and \(b=5\). Then \(a^{2}=10^{2} = 100\). So, \(10\sqrt{5}=\sqrt{10^{2}\times5}=\sqrt{100\times5}=\sqrt{500}\).

Step3: Check other options

• For option B: \(\sqrt{105}\) cannot be simplified to \(10\sqrt{5}\) since \(105 = 3\times5\times7\) and there is no perfect square factor.
• For option C: \(\sqrt{50}=\sqrt{25\times2} = 5\sqrt{2}

eq10\sqrt{5}\)

• For option D: \(\sqrt{15}\) has no perfect square factors and is not equal to \(10\sqrt{5}\)

Answer:

A. \(\sqrt{500}\)