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Question
Evaluate:
`root(3)(27) + root(3)(0.008) + root(3)(0.064)`
Solution
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
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