Question

simplify. assume q is greater than or equal to zero. \\(sqrt{50q^{9}}\\)

Explanation:

Step1: Factor the radicand

We can factor \(50q^9\) into perfect square factors and other factors. First, factor \(50\) as \(25\times2\) and \(q^9\) as \(q^8\times q\) (since \(q^8=(q^4)^2\) is a perfect square and \(q\) is the remaining factor). So, \(\sqrt{50q^9}=\sqrt{25\times2\times q^8\times q}\).

Step2: Use the property of square roots \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\)

Applying the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (where \(a = 25q^8\) and \(b=2q\)), we get \(\sqrt{25q^8\times2q}=\sqrt{25q^8}\times\sqrt{2q}\).

Step3: Simplify \(\sqrt{25q^8}\)

Since \(\sqrt{25}=5\) and \(\sqrt{q^8}=q^4\) (because \((q^4)^2 = q^8\) and \(q\geq0\)), then \(\sqrt{25q^8}=5q^4\).

Step4: Combine the results

Putting it all together, \(\sqrt{25q^8}\times\sqrt{2q}=5q^4\sqrt{2q}\).

Answer:

\(5q^4\sqrt{2q}\)