Question

recall that to simplify a square root, you can write the radicand as a product with factors that are perfect squares. for example: √108 = √(36·3) = 6√3 simplify: √240 16 + √15 16√15 4 + √15 4√15

Explanation:

Step1: Factor the radicand

Find the largest perfect square factor of 240. The factors of 240 are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240. The largest perfect square factor is 16 (since \(16 = 4^2\)). So, we can write \(240\) as \(16\times15\). Thus, \(\sqrt{240}=\sqrt{16\times15}\).

Step2: Use the property of square roots

Recall the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0\) and \(b\geq0\)). Applying this property to \(\sqrt{16\times15}\), we get \(\sqrt{16}\times\sqrt{15}\).

Step3: Simplify \(\sqrt{16}\)

We know that \(\sqrt{16} = 4\) because \(4^2 = 16\). So, substituting this back, we have \(4\times\sqrt{15}=4\sqrt{15}\).

Answer:

\(4\sqrt{15}\) (corresponding to the option with \(4\sqrt{15}\))