17:13a

These kings have one purpose: This clause indicates that the ten kings will be united in how they rule. They will be in agreement. Other ways to translate this clause are:

These ten all have the same purpose (Good News Translation)
-or-
These kings have a single intent (NET Bible)
-or-
These are of one mind (Revised Standard Version)

17:13b

to yield their power and authority to the beast: This clause explains what their united purpose is. The ten kings use their power to serve the beast and obey him. For example:

in putting their strength and their powers at the beast’s disposal (New Jerusalem Bible)
-or-
namely, they will ⌊completely⌋ obey the beast with all their power and authority

power: Here the word power refers to having the ability to do something.

authority: Here the word authority refers to having the right and power to control or rule over someone or something.

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