Question

what is the radical part of both \\(\sqrt3{54}\\) and \\(\sqrt3{128}\\) when the expressions are simplified? \\(\sqrt3{2}\\) 2 \\(\sqrt3{3}\\) \\(\sqrt{2}\\)

Explanation:

Step1: Simplify $\sqrt[3]{54}$

Factor 54: $54 = 27\times2 = 3^3\times2$. So, $\sqrt[3]{54}=\sqrt[3]{3^3\times2}=\sqrt[3]{3^3}\times\sqrt[3]{2}=3\sqrt[3]{2}$.

Step2: Simplify $\sqrt[3]{128}$

Factor 128: $128 = 64\times2 = 4^3\times2$. So, $\sqrt[3]{128}=\sqrt[3]{4^3\times2}=\sqrt[3]{4^3}\times\sqrt[3]{2}=4\sqrt[3]{2}$.

Step3: Identify radical part

From the simplified forms $3\sqrt[3]{2}$ and $4\sqrt[3]{2}$, the radical part (the part with the cube root) is $\sqrt[3]{2}$.

Answer:

$\boldsymbol{\sqrt[3]{2}}$ (the first option: $\sqrt[3]{2}$)