### Which is largest among 2^{1/2}, 3^{1/3}, 8^{1/8} and 9^{1/9}.

**Answer:**

3^{1/3}

**Step by Step Explanation:**

- If we raise each of the given numbers 2
^{1/2}, 3^{1/3}, 8^{1/8}and 9^{1/9}by same power, the largest number will still be found at the same position. - Let's raise each of the given numbers by a number which will make the powers of each number an integer. Such a number will be the LCM of 2, 3, 8, and 9 which is 72.
- Raising each number by a power of 72, we get the following sequence of numbers:

2^{72/2}, 3^{72/3}, 8^{72/8}and 9^{72/9}

2^{36}, 3^{24}, 8^{9}and 9^{8}

2^{36}, 3^{24}, (2^{3})^{9}and (3^{2})^{8}

2^{36}, 3^{24},2^{27}and 3^{16} - We can see that, 2
^{36}> 2^{27}and 3^{24}> 3^{16}

So, we only need to find the larger of 2^{36}and 3^{24},

Or the larger of (2^{3})^{12}and (3^{2})^{12},

Or the larger of 8^{12}and 9^{12}. - Clearly 9
^{12}is larger. The number corresponding to 9^{12}is 3^{1/3}. Thus**3**is the largest of all.^{1/3}