Question

simplify. assume k is greater than or equal to zero. \\(sqrt{45k^{5}}\\)

Explanation:

Step1: Factor the radicand

We can factor \(45k^5\) into \(9\times5\times k^4\times k\). So, \(\sqrt{45k^5}=\sqrt{9\times5\times k^4\times k}\).

Step2: Use the property of square roots

Recall that \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (for \(a\geq0,b\geq0\)) and \(\sqrt{a^2}=a\) (for \(a\geq0\)). We know that \(\sqrt{9} = 3\), \(\sqrt{k^4}=k^2\) (since \(k\geq0\)). Then we have:
\(\sqrt{9\times5\times k^4\times k}=\sqrt{9}\times\sqrt{k^4}\times\sqrt{5k}=3\times k^2\times\sqrt{5k}\)

Step3: Simplify the expression

Multiplying the terms together, we get \(3k^2\sqrt{5k}\).

Answer:

\(3k^2\sqrt{5k}\)