Question

simplify.
\sqrt{250}

Explanation:

Step1: Factor 250 into prime factors

We know that \(250 = 25\times10\), and \(25 = 5^2\), \(10 = 2\times5\). So \(250=5^2\times2\times5 = 5^3\times2\)? Wait, no, let's do it correctly. \(250 = 2\times5\times5\times5=2\times5^3\)? Wait, no, \(5\times5 = 25\), \(25\times10 = 250\), and \(10 = 2\times5\), so \(250=2\times5\times5\times5 = 2\times5^3\)? Wait, no, \(5^2\times10=25\times10 = 250\), and \(10 = 2\times5\), so \(250 = 5^2\times2\times5=5^3\times2\)? Wait, no, actually, \(250 = 2\times5^3\)? Wait, no, \(5^3=125\), \(125\times2 = 250\), yes. But we want to factor out the perfect square. The largest perfect square factor of 250 is \(5^2 = 25\), because \(250\div25 = 10\). So \(250 = 25\times10\), where 25 is a perfect square (\(5^2\)) and 10 is \(2\times5\).

Step2: Apply the square root property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\))

So \(\sqrt{250}=\sqrt{25\times10}\). Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\), we get \(\sqrt{25}\times\sqrt{10}\).

Step3: Simplify \(\sqrt{25}\)

We know that \(\sqrt{25}=5\) because \(5^2 = 25\). So \(\sqrt{25}\times\sqrt{10}=5\sqrt{10}\).

Answer:

\(5\sqrt{10}\)