Question

simplify. assume a is greater than or equal to zero. \\(sqrt{18a^{4}}\\)

Explanation:

Step1: Factor the radicand

We can factor \(18a^4\) as \(9\times2\times a^4\), since \(9\) is a perfect square and \(a^4=(a^2)^2\) is also a perfect square. So we have \(\sqrt{18a^4}=\sqrt{9\times2\times a^4}\).

Step2: Use the property of square roots

The property of square roots states that \(\sqrt{xy}=\sqrt{x}\times\sqrt{y}\) (for \(x\geq0,y\geq0\)) and \(\sqrt{x^2}=x\) (for \(x\geq0\)). Applying these properties, we get \(\sqrt{9\times2\times a^4}=\sqrt{9}\times\sqrt{2}\times\sqrt{a^4}\).
Since \(\sqrt{9} = 3\) and \(\sqrt{a^4}=a^2\) (because \(a\geq0\)), we substitute these values in: \(3\times\sqrt{2}\times a^2\).

Step3: Simplify the expression

Multiplying the terms together, we get \(3a^2\sqrt{2}\).

Answer:

\(3a^2\sqrt{2}\)