AP ICET 2014

$$\left(a^{\frac{1}{z - x}}\right)^{\frac{1}{z - y}}.\left(a^{\frac{1}{x - y}}\right)^{\frac{1}{x - z}}.\left(a^{\frac{1}{y - z}}\right)^{\frac{1}{y - x}} =$$

$$=a^{\left(\frac{1}{\left(x-z\right)\left(y-z\right)}\right)}\cdot a^{\left(\frac{1}{\left(x-y\right)\left(x-z\right)}\right)}\cdot a^{\left(-\frac{1}{\left(y-z\right)\left(x-y\right)}\right)}$$

$$=a^{\left(\frac{1}{\left(x-z\right)\left(y-z\right)}+\frac{1}{\left(x-y\right)\left(x-z\right)}-\frac{1}{\left(y-z\right)\left(x-y\right)}\right)}$$

Adding up all the terms in the powers yields 0 as the answer. 
Hence, a raised to 0 = 1

The RHS of the equation can be modified as $$\left(\sqrt{\ \frac{81}{625}}\right)^{-\frac{\left(a+3\right)}{2}}=\left(\sqrt{\ \frac{3}{5}}\right)^{4\cdot\left(-\frac{\left(a+3\right)}{2}\right)}$$

As the bases are equal on both sides, respective powers can be equated:
a = -2*(a+3)
a = -2

The amount of milk present in the solution at start = (4/5)*35 = 28 liters
Amount of water present at the start = 35 - 28 = 7 liters
Now, one liter water is added and hence, total water present in solution = 8 liters. 
Hence, ratio of milk and water in the mixture now = 28 : 8 = 7 : 2

Let the salaries be 4x and 7x. 
As both spend 80% of their salaries and save 20%, the amounts saved are 0.2*4x = 0.8x and 0.2*7x = 1.4x
The ratios of savings = 0.8:1.4 = 4:7

The RHS of the equation can be re written as $$\left(\sqrt{\ 2}\right)^{\frac{2x^2-2}{2}}=\left(\sqrt{\ 2}\right)^{x^2-1}$$
As the bases in both LHS and RHS are now same, the powers can be equated. 

$$x+5=x^2-1$$
solving this equation gives us x = 3 or x = -2
For the desired value, we get either 8 or 3. Option having 8 is present, and hence, the answer.