Question

simplify. assume b is greater than or equal to zero. \\(sqrt{75b^{10}}\\)

Explanation:

Step1: Factor the radicand

We can factor \(75b^{10}\) as \(25\times3\times b^{10}\), where \(25\) is a perfect square and \(b^{10}=(b^{5})^{2}\) is also a perfect square (since \(b\geq0\)). So we have \(\sqrt{75b^{10}}=\sqrt{25\times3\times b^{10}}\).

Step2: Use the property of square roots

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\)), we can split the square root: \(\sqrt{25\times3\times b^{10}}=\sqrt{25}\times\sqrt{3}\times\sqrt{b^{10}}\).

Step3: Simplify each square root

We know that \(\sqrt{25} = 5\) and \(\sqrt{b^{10}}=b^{5}\) (because \(b\geq0\)). So substituting these values back, we get \(5\times\sqrt{3}\times b^{5}\).

Step4: Rearrange the terms

Rearranging the terms, we have \(5b^{5}\sqrt{3}\).

Answer:

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