Question

Compare the following quantities in ascending order.

\(\sqrt 3,\ \ \ \root 3 \of 5 ,\ \ \ \ \root 4 \of 7 ,\ \ \ \ \root 6 \of 9 ,\ \ \ \ \root {12} \of {11}\)

1. ​\(\root {12} \of {11} < \root 4 \of 7 < \root 6 \of 9 < \root 3 \of 5 < \sqrt 3\)​
2. \(\root {12} \of {11} < \root 6 \of 9 < \root 4 \of 7< \sqrt 3< \root 3 \of 5 \)​
3. \(\root 4 \of 7 < \root 6 \of 9 < \root 3 \of 5 < \sqrt 3 < \root {12} \of {11}\)​​
4. \(\root {12} \of {11} < \root 6 \of 9 < \root 4 \of 7 < \root 3 \of 5 < \sqrt 3\)

Answer 1

Correct Answer - Option 4 : \(\root {12} \of {11} < \root 6 \of 9 < \root 4 \of 7 < \root 3 \of 5 < \sqrt 3\)

Given:

\(\sqrt 3,\ \ \ \root 3 \of 5 ,\ \ \ \ \root 4 \of 7 ,\ \ \ \ \root 6 \of 9 ,\ \ \ \ \root {12} \of {11}\)

Calculations:

We can write the given numbers as

(3)1/2,  (5)1/3 , (7)1/4 , (9)1/6 , (11)1/12

The LCM of (2, 3, 4, 6, 12) = 12

So, we can write the given numbers as ,

(3)6/12,   (5)4/12 , (7)3/12 , (9)2/12 , (11)1/12

⇒ (729)1/12,  (625)1/12,  (343)1/12 ,  (81)1/12, (11)1/12

Now it is easy for us to compare the terms,

(11)1/12 < (81)1/12 < (343)1/12 < (625)1/12 < (729)1/12

The ascending order is \(\root {12} \of {11} < \root 6 \of 9 < \root 4 \of 7 < \root 3 \of 5 < \sqrt 3\) .