Ake was part of the Bournemouth team which got relegated from the Premier League in the recently-completed season, having spent the last three years at the Vitality Stadium on a permanent basis after an initial spell on loan from Chelsea.
In spite of his side's travails at the back, the 25-year-old's performances have long caught the eye and he will move to the Etihad, where he is looking forward to linking up with manager Pep Guardiola.
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"City have been the best side in England over the course of the last decade,” Ake said.
“Coming here is a dream for me. This is a top side full of world-class players. Everywhere you look in this squad there are big names with international pedigree.

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“Pep is a manager admired across the world – what he’s done in the game speaks for itself. The success he’s had is unbelievable and the style of football he plays really appeals to me.
“I know I’m going to have to work hard to get into the side, but that’s what I’m here to do. I’ll do whatever I can to make an impact and help the team win silverware.”
The Netherlands international moved to Chelsea when he was a teenager but spent the majority of his time at Stamford Bridge on various loan spells.
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