Prove that :

$\dfrac{a^{-1}}{a^{-1} + b^{-1}} + \dfrac{a^{-1}}{a^{-1} - b^{-1}} = \dfrac{2b^2}{b^2 - a^2}$

Prove that :

$\dfrac{a + b + c}{a^{-1} b^{-1} + b^{-1}c^{-1} + c^{-1}a^{-1}} = abc$

Find the value of x:

(3 + 4) (3² + 4²) (3⁴ + 4⁴) (3⁸ + 4⁸) (3¹⁶ + 4¹⁶) (3³² + 4³²) = (4^x - 3^x)

Mr. Mohan divided a sum of money into three parts 5^x , 3^y and 2^z and distributed among his three sons Aman, Aryan and Asmit respectively. The product of their share is 12,96,000.

(i) Find the share of Aman.

(ii) Find the sum of shares of Aryan and Asmit.

(iii) Find the value of (x + y + z).

(iv) Find the value of $\Big(\sqrt{x^y}\Big)^{z - 5}$