How many 1 kiloton bombs would need to be exploded to produce 3.86x10^26 Joules, the amount of energy emitted by the Sun in 1 second?

This is a unit conversion problem, and the first step is figuring out how to relate a '1 kiloton bomb' explosion to Joules. To do that, you either need to know or figure out what is meant by '1 kiloton bomb' - it
*doesn't* mean that the bomb weighs a thousand tons ('kilo' is the 'thousand' prefix in the SI system), but rather 'kiloton' (or kt) is shorthand for something like 'thousand tons of TNT explosive equivalent.' Why is this done and just how much energy is in the explosion of a thousand tons of TNT?

Good question! Nuclear weapon blasts release enormous amounts of energy, so a unit with an appropriate scale is useful - can you imagine trying to measure (or compare) a person's height in Plank lengths (http://wolfr.am/10N3d2K) or Astronimcal Units (http://wolfr.am/inG4Ep)? Of course not. Nuclear weapon explosions are usually measured in kilotons or megatons; relatively large nonnuclear explosions (chemical explosives or meteor impacts) are also sometimes measured using units of (kilo/mega)tons of TNT.

As a unit of energy, 1 gram of TNT has been chosen to be equivalent to 1 kilocalorie, or 4184 joules (= 4.184 x 10^{3}). Why? A gram of TNT actually releases around that much energy anyway, and deciding to fix its value as 1 kilocalorie makes all sorts of calculations convenient to do. Anyway, if an explosion equivalent to 1 gram of TNT releases 4184 joules, a kilogram (a thousand grams) of TNT releases a thousand times this much energy - 1000 x 4.184 x 10^{3} = 4.184 x 10^{6} joules. A metric ton of TNT (a thousand kg) releases 1000 times this (4.184 x 10^{9} joules), and a thousand tons of TNT (1 kt), a thousand times
**that** - 4.184 x 10^{12} joules.

Finally, then, to find the number of 1 kt bomb explosions needed to produce 3.86 x 10^{26} joules, divide 3.86 x 10^{26} joules by 4.184 x 10^{12} joules.

(Let me know if any of the math, especially manipulating the exponents in scientific notation is unclear or why division is used in the final step.)