Find a and b.

$12^{\frac{3}{4}}\times \sqrt[a]{12^b}=12^{\frac{3}{2}}\times \sqrt{12^7}\\ 12^{\frac{3}{4}+\frac{b}{a}}=12^{\frac{3}{2}+\frac{7}{2}}\\ \dfrac{3}{4} + \dfrac{b}{a}=5\\ \dfrac{b}{a} = 4\dfrac{1}{4} = \dfrac{17}{4}$

b = any multiple of 17 and a = the corresponding multiple of 4.

That means when b = 2 x 17 = 34, a = 2 x 4 = 8.

i.e. a = 4k, b=17k, where k is a positive integer.

You cannot solve for a and b seperately unless you have 2 equations.

b=17

a=4