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[LOTR] Let's say during the middle of the War of the Ring, Sauron just up and dies. Would Saruman at his peak have been able to conquer Middle Earth?

Did Saruman, before his army and Isengard infrastructure get completely wrecked by a bunch of trees, possess the strength to militarily take over the Free Peoples of Middle Earth?

Would his powers of persuasion have helped him to do so, or were they already deteriorating by the time of his war against the Rohirrim?
gpt
No, Saruman could not have conquered all of Middle Earth. He wasn't even able to conquer Rohan. 

He has mass-production technology capable of producing both Uruk-hai and the weapons with which to arm them, but he doesn't have enough raw resources or a sufficiently large industrial base to create and supply an army large enough to conquer all of Middle Earth. 

At the Battle of the Hornburg, he was able to deploy 10,000 Uruk-hai, 5,000 Orcs and another 2000 Dunlandings (the tribal hill-folk that he persuaded to rise up against Rohan and raid villages in its territory). 17,000 soldiers in total. 

If we assume that he was expecting to wipe Rohan out in one fell swoop and wasn't expecting any sort of immediate retaliation, then we can conclude that those 17,000 troops represented the majority of the soldiers available to him.

He clearly kept enough Orcs to keep running his production machinery and a small garrison force to ward off random raiders, but not a full-on "reserve army", because he ends up trapped in his own tower once the Ents rise up and wipe out his garrison. If he had more troops to call upon, he would have done so at that moment in order to free himself. 

Someone who wants to conquer all of Middle Earth needs a force capable of destroying the combined forces of Gondor, Rohan, the Dwarf clans, and the few small groups of Elves that decide to stay in Middle Earth and fight alongside the younger races instead of going to the Undying Lands. 

17,000 soldiers isn't enough to overcome all those armies. Sauron had 45,000 soldiers at the Battle of the Pelannor Fields, and that wasn't enough to take Minas Tirith. 

At the Battle of the Black Gate, which was happening at the very moment that Frodo and Sam were making their way up Mount Doom to throw in the ring, Sauron deployed a combined force of 60,000 soldiers to face Aragorn's force of 10,000. 

Saruman was a potent regional-level threat to Rohan (or he would have been if he hadn't chosen to fight them at the one location where they actually stood a chance of winning, although even then he was on track to win a Pyrrhic victory through strength of numbers before Gandalf returned with the Rohirrim), but he would get absolutely crushed if he attempted to take on all of Middle Earth.