Question

show work to simplify
$12\sqrt{75} - 4\sqrt{12}$
remember to first break down the radicands to get similar radicands.

Explanation:

Step1: Simplify \(\sqrt{75}\)

We break down 75 into factors where one is a perfect square. \(75 = 25\times3\), so \(\sqrt{75}=\sqrt{25\times3}\). Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\)), we get \(\sqrt{25\times3}=\sqrt{25}\times\sqrt{3}=5\sqrt{3}\). Then \(12\sqrt{75}=12\times5\sqrt{3}=60\sqrt{3}\).

Step2: Simplify \(\sqrt{12}\)

Break down 12 into factors with a perfect square. \(12 = 4\times3\), so \(\sqrt{12}=\sqrt{4\times3}\). Using the same square - root property, \(\sqrt{4\times3}=\sqrt{4}\times\sqrt{3}=2\sqrt{3}\). Then \(4\sqrt{12}=4\times2\sqrt{3}=8\sqrt{3}\).

Step3: Subtract the two simplified terms

Now we have \(12\sqrt{75}-4\sqrt{12}=60\sqrt{3}-8\sqrt{3}\). Since the radicands are the same, we can subtract the coefficients: \((60 - 8)\sqrt{3}=52\sqrt{3}\).

Answer:

\(52\sqrt{3}\)