Question

4 simplify. \\(\dfrac{\sqrt3{270x^{20}}}{\sqrt3{5x}}\\) \\(\boldsymbol{\text{a}}\\) \\(2x\sqrt3{3x^6}\\) \\(\boldsymbol{\text{b}}\\) \\(\sqrt3{135x^{19}}\\) \\(\boldsymbol{\text{c}}\\) \\(3x^6\sqrt{135x}\\) \\(\boldsymbol{\text{d}}\\) \\(3x^6\sqrt3{2x}\\)

Explanation:

Step1: Use Quotient Rule for Radicals

The quotient rule for cube roots states that \(\frac{\sqrt[3]{a}}{\sqrt[3]{b}}=\sqrt[3]{\frac{a}{b}}\). So we can combine the two cube roots into one: \(\sqrt[3]{\frac{270x^{20}}{5x}}\)

Step2: Simplify the Fraction Inside the Radical

Simplify the coefficient and the variable separately. For the coefficient: \(\frac{270}{5} = 54\). For the variable: \(\frac{x^{20}}{x}=x^{20 - 1}=x^{19}\). So now we have \(\sqrt[3]{54x^{19}}\)

Step3: Factor the Radicand into Perfect Cubes and Remainders

Factor \(54\) as \(27\times2\) (since \(27 = 3^3\)) and \(x^{19}\) as \(x^{18}\times x\) (since \(x^{18}=(x^{6})^3\)). So \(\sqrt[3]{27\times2\times x^{18}\times x}\)

Step4: Apply the Product Rule for Radicals

The product rule for cube roots states that \(\sqrt[3]{ab}=\sqrt[3]{a}\times\sqrt[3]{b}\). So we can split the radical: \(\sqrt[3]{27x^{18}}\times\sqrt[3]{2x}\)

Step5: Simplify the Perfect Cube Radical

\(\sqrt[3]{27x^{18}}=\sqrt[3]{3^3\times(x^{6})^3}=3x^{6}\). So putting it together, we get \(3x^{6}\sqrt[3]{2x}\)

Answer:

D. \(3x^{6}\sqrt[3]{2x}\)