Simple equation

$5^{x} + 5^{x+3} = 15750$

$5^{x} + 5^{x}5^{3} = 15750$

$5^{x}(1 + 5^{3}) = 15750$

$5^{x}(1+125)=15750$

$5^{x} \times 126 = 15750$

$5^{x} = \frac {15750}{126}$

$5^{x} = 125$

$5^{x} = 5^{3}$

$\therefore \boxed {x=3}$

Simple, let $5^x=y$

Using the law of exponents, we arrange express the equation as $y+125y=15750$ and $y=125$ , that means that $\boxed{x=3}$

5^x + 5^x+3 = 15750

5^x + 5^x+3 = 15625 + 125

5^x + 5^x+3 = 5^6 + 5^3

(5^x) + (5^x+3) = (5^3) + (5^3+3)

Making the comparison between the equallity, we can conclude: x = 3