I want to make technology that can explode the whole Earth. The problem is that antimatter explosions and nuclear explosions just don't have the juice needed to blow up the planet. Conventional explosions are just pathetic in terms of planet kabooms; they don't even scratch the surface. What explosion would be powerful enough that only one to twenty uses of the explosions are needed to blow up the Earth? I specifically want an explosive (no pun intended) superpower; in my world I want to blow up the Earth with a giant explosion.
Please note that this answer was provided before Arachian, with my encouragement, attached the Science-Based tag.
A fascinating article about supernova ASASSN-15lh suggests something absolutely amazing...
In this study, we examine the maximum luminosity possible for pair-instability supernovae (PISNe). We also examine how much radioactive nickel is needed for ASASSN-15lh, if it is considered to be purely nickel-powered. We describe our toy models in Section 2, discuss the light curves and derive a relation between nickel mass and resulting supernova peak luminosity in Sections 3 and 4.
To make a long and complicated story short, the article suggests the supernova may have been caused by nickel.
The bad news? They think it took a whomping lot of nickel to make the supernova do what it did.
The good news? All you're looking for is suspension of disbelief. And if nature can blow up stars using nickel, then your mad scientist can blow up a planet with a liquid nickel-iron core.
In other words, in your story he proved the scientists were right about ASASSN-15lh... and he weaponized their work.
He figured out how to turn the core of the planet into a nuclear bomb. The basics? He created a super-shaped-charge bomb and figured out how to get that shaped bomb to the inner core of the planet, then set it off. It forced enough of the nickel core to critical mass that it started a chain reaction.
You didn't tag your question with any of the Science tags. Therefore, please note that I'm suggesting a science-fiction solution because I can't imagine a practical science-based solution that's better. And if you aren't using a mad scientist, I apologize, but this is definitely a question worthy of Dr. Farnsworth.
Any method of "blowing up the earth", needs to at least move all the material from the earth out of the earth's gravity-well. This places a lower bound on the amount of energy needed, equal to the gravitational binding energy of the earth. This is roughly equal to 2.49 x 10^32 Joules. This is an unreasonable amount of energy, but let's consider some options.
The energy is about a quadrillion times as much as the yield of the most powerful nuclear bomb ever created. The practical maximum yield-to-weight ratio has been estimated to 25 TJ/kg (the Taylor limit), meaning our planet-killer-bomb would need to weigh at least eight quadrillion metric tons.
An antimatter-bomb would react matter with antimatter to produce energy according to E=mc^2. A perfectly efficient antimatter-bomb would need to react 1.1 trillion metric tons of matter with the same amount of antimatter, to yield the required energy. Again, somewhat unwieldy.
Maybe a stellar death-ray could work? But, no. The energy is equivalent to roughly a week of the total output of the sun, so even a dyson-sphere-like device, capturing the total output of the sun, would need to cook the planet for a week in order to slowly ablate it away. Not an explosion by any means.
The only reasonable science-based way to obtain this much energy that I can think of would be to somehow redirect another planet, and send it on a collision course with the earth. In fact, the only number I've seen while researching this answer which is bigger than the gravitational binding energy of the earth, is the earths kinetic energy in it's orbit. This would allow as much time as needed to move the other planet, but still result in a sudden release of energy upon the earth, which might then be classified as an explosion.
Rapidly spinning miniature black hole
Note also that the only limit to the amount of power generated is the size of the black hole.
Scale up and you can blow up the planet.
Also, note that a black hole made out of the mass of a mountain has the mass of a mountain after it becomes a black hole. It doesn’t get heavier when it becomes a black hole. So if you drop it into the earth it won’t swallow the earth by increasing gravity that much and it won’t absorb too much material on its way down.