Question

Evaluate:

`root(3)(27) + root(3)(0.008) + root(3)(0.064)`

Answer 1

We have, `root(3)(27) + root(3)(0.008) + root(3)(0.064)`

27 = 3 × 3 × 3 = 3³

`\implies root(3)(27) = 3`

`0.008 = 8/1000`

And 8 = 2 × 2 × 2 = 2³

`\implies root(3)(8) = 2`

Also, 1000 = 2 × 2 × 2 × 5 × 5 × 5 = (2 × 5)³ 

`\implies root(3)(1000) = 2 xx 5 = 10`

∴ `root(3)(0.008) = (root(3)(8))/(root(3)(1000)) = 2/10 = 0.2`

`0.064 = 64/1000`

And 64 = 2 × 2 × 2 × 2 × 2 × 2 = (2 × 2)³

`\implies root(3)(64) = 2 xx 2 = 4`

And 1000 = 2 × 2 × 2 × 5 × 5 × 5 = (2 × 5)³ 

`\implies root(3)(1000) = 2 xx 5 = 10`

∴ `root(3)(0.064) = (root(3)(64))/(root(3)(1000)) = 4/10 = 0.4`

Thus, `root(3)(27) + root(3)(0.008) + root(3)(0.064)`

= 3 + 0.2 + 0.4

= 3.6