Question

simplify. assume ( b ) is greater than or equal to zero. (sqrt{75b^{3}})

Explanation:

Step1: Factor the radicand

Factor \(75b^3\) into perfect square factors and remaining factors. We know that \(75 = 25\times3\) and \(b^3=b^2\times b\). So, \(75b^3 = 25\times3\times b^2\times b\).

Step2: Apply the square - root property

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\)), we have \(\sqrt{75b^3}=\sqrt{25\times b^2\times3\times b}\).
Since \(\sqrt{25}=5\) and \(\sqrt{b^2}=b\) (because \(b\geq0\)), we can rewrite the expression as \(\sqrt{25}\times\sqrt{b^2}\times\sqrt{3b}\).
Substituting the values of \(\sqrt{25}\) and \(\sqrt{b^2}\), we get \(5\times b\times\sqrt{3b}\).

Step3: Simplify the expression

Simplifying \(5\times b\times\sqrt{3b}\), we have \(5b\sqrt{3b}\).

Answer:

\(5b\sqrt{3b}\)