Arrange the following surds in increasing order :
(A) $$\sqrt[3]{2}$$
(B) $$\sqrt{3}$$
(C) $$\sqrt[4]{5}$$
(D) $$\sqrt[6]{7}$$
Choose the correct answer from the options given below :

We are given the surds, $$\sqrt[3]{2}$$, $$\sqrt{3}$$, $$\sqrt[4]{5}$$ and $$\sqrt[6]{7}$$.

The values can be written as, $$2^{\frac{1}{3}},\ 3^{\frac{1}{2}},\ 5^{\frac{1}{4}},\ 7^{\frac{1}{6}}$$ which can be rewritten as,

$$2^{\frac{8}{24}},\ 3^{\frac{12}{24}},\ 5^{\frac{6}{24}},\ 7^{\frac{4}{24}}$$.

In the above values, comparing the values of $$2^8,\ 3^{12},\ 5^6,\ 7^4$$ is enough and they can be rewritten as,

$$\left(2^4\right)^2,\ \left(3^6\right)^2,\ \left(5^3\right)^2,\ \left(7^2\right)^2\ =\ $$ $$16^2,\ 729^2,\ 125^2,\ 49^2$$

We can see that the value of A < D < C < B.

Therefore, the correct answer is option D.