Question

simplify. assume n is greater than or equal to zero.\\(\sqrt{12n^{8}}\\)

Explanation:

Step1: Factor the radicand

We can factor \(12n^8\) into \(4\times3\times n^8\), where \(4\) is a perfect square. So we have \(\sqrt{4\times3\times n^8}\).

Step2: Use the property of square roots \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\))

Applying this property, we get \(\sqrt{4}\times\sqrt{3}\times\sqrt{n^8}\).

Step3: Simplify each square root

We know that \(\sqrt{4} = 2\), and for \(\sqrt{n^8}\), since \(n\geq0\), using the property \(\sqrt{x^m}=x^{\frac{m}{2}}\) (for \(x\geq0\)), we have \(n^{\frac{8}{2}}=n^4\). So putting it together, we get \(2\times n^4\times\sqrt{3}\).

Step4: Rewrite the expression

The simplified form is \(2n^4\sqrt{3}\).

Answer:

\(2n^{4}\sqrt{3}\)