Question

simplify. assume a is greater than or equal to zero. \\(sqrt{75a^{6}}\\)

Explanation:

Step1: Factor the radicand

We can factor \(75a^6\) into perfect square factors and other factors. First, factor \(75\) as \(25\times3\) and \(a^6\) as \((a^3)^2\) (since \((a^m)^n=a^{mn}\), so \(a^6 = a^{3\times2}=(a^3)^2\)). So we have \(\sqrt{75a^6}=\sqrt{25\times3\times(a^3)^2}\).

Step2: Use the property of square roots \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\))

We can split the square root as \(\sqrt{25}\times\sqrt{3}\times\sqrt{(a^3)^2}\).

Step3: Simplify each square root

We know that \(\sqrt{25} = 5\) (since \(5^2 = 25\)), \(\sqrt{(a^3)^2}=a^3\) (because \(a\geq0\), so the square root of a square is the non - negative number itself), and \(\sqrt{3}\) remains as it is. Then we multiply these results together: \(5\times a^3\times\sqrt{3}=5a^3\sqrt{3}\).

Answer:

\(5a^3\sqrt{3}\)