Question

simplify \\(\sqrt3{5x} \cdot \sqrt3{25x^2}\\) completely.\
\\(25x^3\\) \\(25x\\) \\(5x\\)\
\\(5x^3\\)

Explanation:

Step1: Use product rule for radicals

The product rule for cube roots states that \(\sqrt[3]{a} \cdot \sqrt[3]{b}=\sqrt[3]{ab}\). So, \(\sqrt[3]{5x} \cdot \sqrt[3]{25x^{2}}=\sqrt[3]{(5x)(25x^{2})}\).

Step2: Multiply the terms inside the cube root

Multiply the coefficients and the variables separately. For coefficients: \(5\times25 = 125\). For variables: \(x\times x^{2}=x^{1 + 2}=x^{3}\). So we have \(\sqrt[3]{125x^{3}}\).

Step3: Simplify the cube root

We know that \(\sqrt[3]{125}=5\) because \(5^{3}=125\) and \(\sqrt[3]{x^{3}} = x\) because \((x)^{3}=x^{3}\). So \(\sqrt[3]{125x^{3}}=5x\).

Answer:

\(5x\) (the option with "5x" as the text)