Question

simplify. assume a is greater than or equal to zero. \\(sqrt{18a^{5}}\\)

Explanation:

Step1: Factor the radicand

Factor \(18a^5\) into perfect - square factors and non - perfect - square factors. We know that \(18 = 9\times2\) and \(a^5=a^{4 + 1}=a^4\times a\). So, \(\sqrt{18a^5}=\sqrt{9\times2\times a^4\times a}\).

Step2: Use the property of square roots \(\sqrt{xy}=\sqrt{x}\cdot\sqrt{y}\) (\(x\geq0,y\geq0\))

According to the property \(\sqrt{xy}=\sqrt{x}\cdot\sqrt{y}\), we can rewrite \(\sqrt{9\times2\times a^4\times a}\) as \(\sqrt{9}\times\sqrt{a^4}\times\sqrt{2a}\).

Step3: Simplify the perfect - square roots

We know that \(\sqrt{9} = 3\) and \(\sqrt{a^4}=a^{4\div2}=a^2\) (since \(a\geq0\)). So, \(\sqrt{9}\times\sqrt{a^4}\times\sqrt{2a}=3\times a^2\times\sqrt{2a}\).

Step4: Combine the terms

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

Answer:

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