Question

Classify the following in to like surds:
$\sqrt[3]{32000},\sqrt[3]{3686},\sqrt[3]{3648},\sqrt[3]{3375},\sqrt[3]{3128},\sqrt[3]{324}$

Answer 1

The given cube roots $\sqrt[3]{32000},\sqrt[3]{3686},\sqrt[3]{3648},\sqrt[3]{3375},\sqrt[3]{3128},\sqrt[3]{324}$ can be rewritten as follows:

$\sqrt[3]{32000},\sqrt[3]{3686},\sqrt[3]{3648},\sqrt[3]{3375},\sqrt[3]{3128},\sqrt[3]{324}=\sqrt[3]{32\times 2\times 2\times 2\times 5\times 5\times 5},\sqrt[3]{37\times 7\times 7\times 2},\sqrt[3]{32\times 2\times 2\times 3\times 3\times 3\times 3},\sqrt[3]{35\times 5\times 5\times 3},\sqrt[3]{34\times 4\times 4\times 2},$

$\sqrt[3]{32\times 2\times 2\times 3}=\sqrt[3]{32^4\times 5^3},\sqrt[3]{37^3\times 2},\sqrt[3]{32^3\times 3^4},\sqrt[3]{35^3\times 3},\sqrt[3]{34^3\times 2},\sqrt[3]{32^3\times 3}=\sqrt[3]{32(2^3\times 5^3)},\sqrt[3]{37^3\times 2},\sqrt[3]{33(2^3\times 3^3)},\sqrt[3]{35^3\times 3},\sqrt[3]{34^3\times 2},\sqrt[3]{32^3\times 3}$

$=\sqrt[3]{32\times {10}^3},\sqrt[3]{37^3\times 2},\sqrt[3]{33\times 6^3},\sqrt[3]{35^3\times 3},\sqrt[3]{34^3\times 2},\sqrt[3]{32^3\times 3}=10\sqrt[3]{32},7\sqrt[3]{32},6\sqrt[3]{33},5\sqrt[3]{33},4\sqrt[3]{32},2\sqrt[3]{33}=\{4\sqrt[3]{32},7\sqrt[3]{32},10\sqrt[3]{32}\},\{2\sqrt[3]{33},5\sqrt[3]{33},6\sqrt[3]{33}\}=\{\sqrt[3]{3128},\sqrt[3]{3686},\sqrt[3]{32000}\},\{\sqrt[3]{324},\sqrt[3]{3375},\sqrt[3]{3648}\}$

Hence, the required surds are $\{\sqrt[3]{3128},\sqrt[3]{3686},\sqrt[3]{32000}\}$ and $\{\sqrt[3]{324},\sqrt[3]{3375},\sqrt[3]{3648}\}$.