Question

simplify. assume ( r ) is greater than or equal to zero. (sqrt{18r^{8}})

Explanation:

Step1: Factor the radicand

We can factor \(18r^8\) as \(9\times2\times r^8\), where \(9\) is a perfect square and \(r^8=(r^4)^2\) is also a perfect square. So, \(\sqrt{18r^8}=\sqrt{9\times2\times r^8}\).

Step2: Use the property of square roots

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (where \(a\geq0\) and \(b\geq0\)), we can split the square root: \(\sqrt{9\times2\times r^8}=\sqrt{9}\times\sqrt{2}\times\sqrt{r^8}\).

Step3: Simplify each square root

We know that \(\sqrt{9} = 3\) and \(\sqrt{r^8}=r^4\) (since \(r\geq0\), the square root of \(r^8\) is the non - negative square root). So, substituting these values in, we get \(3\times\sqrt{2}\times r^4 = 3r^4\sqrt{2}\).

Answer:

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