Question 7

Which of the following is TRUE?
$$I. \frac{1}{\sqrt[3]{12}} > \frac{1}{\sqrt[4]{29}} > \frac{1}{\sqrt{5}}$$

$$II. \frac{1}{\sqrt[4]{29}} > \frac{1}{\sqrt[3]{12}} > \frac{1}{\sqrt{5}}$$

$$III. \frac{1}{\sqrt{5}} > \frac{1}{\sqrt[3]{12}} > \frac{1}{\sqrt[4]{29}}$$

$$IV.  \frac{1}{\sqrt{5}} > \frac{1}{\sqrt[4]{29}} > \frac{1}{\sqrt[3]{12}}$$

Solution

$$\frac{1}{\sqrt[3]{12}} , \frac{1}{\sqrt[4]{29}} , \frac{1}{\sqrt{5}}$$

Here if we write $$\frac{1}{\sqrt{5}}$$ in terms of fourth root we have $$\frac{1}{\sqrt[4]{25}}$$

So  $$\frac{1}{\sqrt[4]{29}} >  \frac{1}{\sqrt[4]{5}}$$

$$\sqrt[3]{12}$$=$$\sqrt[12]{144*144}$$

$$\frac{1}{\sqrt[4]{29}}$$=$$\frac{1}{\sqrt[12]{29*29*29}}$$

Therefore $${\sqrt[4]{29}}$$>$$\sqrt[3]{12}$$
As these are given in the denominators order will be reversed.
$$\frac{1}{\sqrt{5}} > \frac{1}{\sqrt[3]{12}} > \frac{1}{\sqrt[4]{29}}$$


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