How exactly does the regieleki/regidrago ability work?

Question

I understand that say that multiplies the attacking stat but, can someone tell me an example?

Answer 1

They power up Electric and Dragon type moves by 50%
Transistor and Dragon's Maw both power up Dragon type or Electric type moves by 50%, or 1.5x.
Transistor powers up Electric type moves by 50%, while Dragon's Maw powers up Dragon type moves by 50%.

Sources: https://bulbapedia.bulbagarden.net/wiki/Transistor_(Ability) and https://bulbapedia.bulbagarden.net/wiki/Dragon's_Maw_(Ability)

Comments

Your sources say they power up electric or dragon attacks, not STAB attacks.

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But they are STAB to Regieleki and Regidrago.

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Those abilities can power up non-STAB moves or not power up STAB moves if they get copied by another Pokemon or if Regieleki or Regidrago change their types.

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You have a point.
Like when a pokemon Traces, for example, right?

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Yes. Trace can trace both of these abilities.

Answer 2

Transistor and dragon's maw don't raise stats. They power up certain moves by 3/2 times.
source
source
For example, thunderbolt is an electric move with 90 base power.
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If a Pokemon with transistor uses thunderbolt, then transistor activates because thunderbolt is an electric attack with a base power, and transistor multiplies thunderbolt's power by 3/2. The resulting power is 90 * 3 / 2 = 135.