Negative exponents under square root sign

(5X10^-4)(3X10^-10) 

\(\sqrt{(5*10^{-4})(3*10^{-10}) }\\ =\sqrt{5*10^{-4}*3*10^{-10} }\\ =\sqrt{5*3*10^{-4}*10^{-10} }\\ =\sqrt{15*10^{-4+-10} }\\ =\sqrt{15*10^{-14} }\\ =\sqrt{15}*\sqrt{10^{-14} }\\ =\sqrt{15}*10^{(-14*\frac{1}{2}) }\\ =\sqrt{15}*10^{(-7) }\\ =\sqrt{15}*\frac{1}{10^{(+7) }}\\ =\frac{\sqrt{15}}{10^7}\\\)

sqrt(15)/10^7 = 0.0000003872983346

We have (5X10^-4)(3X10^-10) under a square root sign (and then further calculations to complete the formula).  We are getting the correct answer but with the decimal point way off.  What do I need to know about negative exponents under square root signs? 

\(\sqrt{{(5\times10^{-4})}\times(3\times10^{-10})}\)

\(=\sqrt{0.0005\times 0.0000000003}\)

\(=\sqrt{0.000 000 000 00015}\)

\(=3.60555127546\times 10^{-7}\)

!

sqrt[(5 x 10^-4)(3x10^-10)]  = sqrt (15 x 10^-14)  = sqrt(15)  x 10^-7 =

0.0000003872983346 = 3.872983346e-7

thanks for the detailed answers. but asinus's answer does not agree with the others. why not?

atdhvaannkcse!