Question

simplify. assume that all variables represent positive real numbers.
\sqrt3{64a^8b^5}
\bigcirc\\ 4ab\sqrt3{a^3b^3}
\bigcirc\\ 4ab\sqrt3{a^2b^2}
\bigcirc\\ 4\sqrt3{a^2b^2}
\bigcirc\\ 4a^2b\sqrt3{a^2b^2}

Explanation:

Step1: Factor the radicand

We can express \(64a^{8}b^{5}\) as a product of perfect cubes and the remaining factors. We know that \(64 = 4^{3}\), \(a^{8}=a^{6}\cdot a^{2}=(a^{2})^{3}\cdot a^{2}\), and \(b^{5}=b^{3}\cdot b^{2}\). So, \(\sqrt[3]{64a^{8}b^{5}}=\sqrt[3]{4^{3}\cdot(a^{2})^{3}\cdot b^{3}\cdot a^{2}\cdot b^{2}}\).

Step2: Apply the cube - root property \(\sqrt[3]{xyz}=\sqrt[3]{x}\cdot\sqrt[3]{y}\cdot\sqrt[3]{z}\)

Using the property of cube roots \(\sqrt[3]{abc}=\sqrt[3]{a}\cdot\sqrt[3]{b}\cdot\sqrt[3]{c}\) (where \(a = 4^{3}\), \(b=(a^{2})^{3}\), \(c = b^{3}\), and the remaining part is \(a^{2}b^{2}\)), we get:
\(\sqrt[3]{4^{3}\cdot(a^{2})^{3}\cdot b^{3}\cdot a^{2}\cdot b^{2}}=\sqrt[3]{4^{3}}\cdot\sqrt[3]{(a^{2})^{3}}\cdot\sqrt[3]{b^{3}}\cdot\sqrt[3]{a^{2}b^{2}}\)
Since \(\sqrt[3]{4^{3}} = 4\), \(\sqrt[3]{(a^{2})^{3}}=a^{2}\), and \(\sqrt[3]{b^{3}} = b\), the expression becomes \(4a^{2}b\sqrt[3]{a^{2}b^{2}}\)

Answer:

\(4a^{2}b\sqrt[3]{a^{2}b^{2}}\) (the last option: \(4a^{2}b\sqrt[3]{a^{2}b^{2}}\))