Which of the given options shows the correct order?

We have to compare:

∛2, ⁶√3, ⁹√4 and ¹⁸√12

Step 1: Find a common root index

The root indices are:

3, 6, 9 and 18

Their LCM is:

LCM(3, 6, 9, 18) = 18

Therefore, convert every expression into an 18th root.

Step 2: Convert each surd
First expression

∛2 = ¹⁸√(2⁶)

2⁶ = 64

Therefore:

∛2 = ¹⁸√64

Second expression

⁶√3 = ¹⁸√(3³)

3³ = 27

Therefore:

⁶√3 = ¹⁸√27

Third expression

⁹√4 = ¹⁸√(4²)

4² = 16

Therefore:

⁹√4 = ¹⁸√16

Fourth expression

¹⁸√12 already has root index 18:

¹⁸√12 = ¹⁸√12

Step 3: Compare the radicands

All expressions now have the same root index. Therefore, compare the numbers inside the roots:

64 > 27 > 16 > 12

Hence:

¹⁸√64 > ¹⁸√27 > ¹⁸√16 > ¹⁸√12

Returning to the original expressions:

∛2 > ⁶√3 > ⁹√4 > ¹⁸√12