Question

simplify. assume m is greater than or equal to zero.\\(\sqrt{27m^{4}}\\)

Explanation:

Step1: Factor the radicand

We can factor \(27m^4\) into perfect square factors and other factors. Recall that \(27 = 9\times3\) and \(m^4=(m^2)^2\). So we can rewrite the radicand as:
\(27m^4 = 9\times3\times(m^2)^2\)

Step2: Apply the square - root property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\))

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\), we have:
\(\sqrt{27m^4}=\sqrt{9\times3\times(m^2)^2}=\sqrt{9}\times\sqrt{(m^2)^2}\times\sqrt{3}\)

Step3: Simplify each square - root term

We know that \(\sqrt{9} = 3\) and \(\sqrt{(m^2)^2}=m^2\) (since \(m\geq0\), the square - root of \(m^4\) is \(m^2\)). Substituting these values in, we get:
\(\sqrt{9}\times\sqrt{(m^2)^2}\times\sqrt{3}=3\times m^2\times\sqrt{3}\)

Step4: Combine the terms

Combining the terms, we have \(3m^2\sqrt{3}\)

Answer:

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