Question

simplify \\(\sqrt3{-64x^5}\\). the simplified expression is \\(\square\\).

Explanation:

Step1: Analyze the cube root of -64

We know that \(\sqrt[3]{-64}\) can be simplified because \((-4)^3=-64\), so \(\sqrt[3]{-64} = - 4\).

Step2: Analyze the cube root of \(x^{5}\)

We can rewrite \(x^{5}\) as \(x^{3}\cdot x^{2}\). Then \(\sqrt[3]{x^{5}}=\sqrt[3]{x^{3}\cdot x^{2}}\). Using the property of cube roots \(\sqrt[3]{ab}=\sqrt[3]{a}\cdot\sqrt[3]{b}\) (where \(a = x^{3}\) and \(b=x^{2}\)), we get \(\sqrt[3]{x^{3}}\cdot\sqrt[3]{x^{2}}=x\sqrt[3]{x^{2}}\).

Step3: Combine the two parts

Now, \(\sqrt[3]{-64x^{5}}=\sqrt[3]{-64}\cdot\sqrt[3]{x^{5}}\). Substituting the values we found in Step 1 and Step 2, we have \(-4\cdot x\sqrt[3]{x^{2}}=-4x\sqrt[3]{x^{2}}\).

Answer:

\(-4x\sqrt[3]{x^{2}}\)