Question : Which of the following is TRUE?
I. $\frac{1}{\sqrt[3]{12}}>\frac{1}{\sqrt[4]{29}}>\frac{1}{\sqrt5}$
II. $\frac{1}{\sqrt[4]{29}}>\frac{1}{\sqrt[3]{12}}>\frac{1}{\sqrt5}$
III. $\frac{1}{\sqrt5}>\frac{1}{\sqrt[3]{12}}>\frac{1}{\sqrt[4]{29}}$
IV. $\frac{1}{\sqrt5}>\frac{1}{\sqrt[4]{29}}>\frac{1}{\sqrt[3]{12}}$
Option 1: Only I
Option 2: Only II
Option 3: Only III
Option 4: Only IV
Correct Answer: Only III
Solution :
Calculating the values: (taking the 12
th
power of each term)
$\frac{1}{\sqrt[3]{12}} \rightarrow \frac{1}{12^4} = \frac{1}{20736}$
$\frac{1}{\sqrt[4]{29}} \rightarrow \frac{1}{29^3} =\frac{1}{24389}$
$\frac{1}{\sqrt5} \rightarrow \frac{1}{5^6} = \frac{1}{15625}$
In fractions, the fraction value is low for fractions with large denominators.
Comparing these values, we get:
$\frac{1}{\sqrt5} >\frac{1}{\sqrt[3]{12}} > \frac{1}{\sqrt[4]{29}}$
Thus, statement III is correct.
Hence, the correct answer is 'Only III'.
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