Question

Greatest among $\sqrt[6]{67},\sqrt[4]{45},\sqrt[5]{56},\sqrt[6]{68}$ is:

• A. $\sqrt[6]{67}$
• B. $\sqrt[4]{45}$
• C. $\sqrt[5]{56}$
• D. $\sqrt[6]{68}$

Answer 1

The correct option is B $\sqrt[4]{45}$

Given, $\sqrt[6]{67},\sqrt[4]{45},\sqrt[5]{56},\sqrt[6]{68}$.

LCM of $6,4,5,6$ is $120$.

So we can write above irrational numbers as

$\sqrt[120]{1207^{20}},\sqrt[120]{1205^{30}},\sqrt[120]{1206^{24}},\sqrt[120]{1208^{20}}$.

Since $5<6<7<8$,

$⟹5^{20}<6^{20}<7^{20}<8^{20}$.

Then, $5^{30-20}=5^{10}$,

$6^{24-20}=6^4$,

$7^{20-20}=7^0=1$

and $8^{20-20}=8^0=1$.

$⟹5^{10}>6^4>1$.

Here, $\sqrt[4]{45}$ is the greatest.