If you cast a spell "without paying its mana cost," you can't choose to cast it for any alternative costs. You can, however, pay additional costs, such as kicker costs. If the spell has any mandatory additional costs, those must be paid to cast it.

If the revealed card has {X} in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

If an opponent casts a spell with {X} in its mana cost, use the value of X that was chosen when it was cast to determine its mana value.

In some unusual cases, the card you reveal from the top of your library may not have the same mana value as the spell your opponent cast, but you can still cast that card because the resulting spell does have the same mana value. For example, if an opponent casts a spell with mana value 4 and you reveal Frogmyr Enforcer (a prototype card with mana cost {7} and prototype cost {3}{R}), you can cast Frogmyr Enforcer as a prototyped spell, as that spell will have a mana value of 4.

The mana value of a split card not on the stack or a fused split spell on the stack is determined by the combined mana cost of its two halves. Otherwise, while a split card is on the stack, its mana value is determined by the mana cost of the half that was chosen to be cast.

A spell's mana value is determined only by its mana cost. Ignore any alternative costs, additional costs, cost increases, or cost reductions.

You choose whether or not to cast the exiled card as Powerbalance's triggered ability resolves. If you do, you do so as part of the resolution of that ability. You can't wait to cast it later in the turn. Timing restrictions based on the card's type are ignored.