Can I use a 9.6A power supply with a 10.83A 360 Slim ?

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It is generally not recommended to use a power supply with lower amperage than what is specified for the device. The power supply amperage rating indicates the maximum amount of current it can provide, while the device's rating indicates its maximum current consumption.

If you use a 9.6A power supply with a 10.83A Xbox 360 Slim, the power supply may not be able to deliver enough current to meet the console's requirements. This can potentially lead to instability, performance issues, or even damage to the power supply or the console itself.

It is best to use a power supply that matches or exceeds the amperage rating specified for the device to ensure proper and safe operation. If you have a 10.83A Xbox 360 Slim, it would be advisable to use a power supply with a similar or higher amperage rating to ensure sufficient power delivery.

Read this, it should help you whether you can or not. https://support.xbox.com/en-GB/help/xbox-360/console/power-supply

but here is what is written at xbox.com: https://support.xbox.com/en-GB/help/xbox-360/console/power-supply

• The 120-watt PSU and 135-watt PSU are interchangeable, and you can use both with the Xbox 360 S and Xbox 360 E consoles.
  

Placed an order in Amazon to resolve this issue

XFUNY (TM) 12V 9.6A AC Adapter Charger Power Supply for Microsoft XBOX 360 E Console

I cannot say for sure on those part numbers, but for the most part, as long as the voltage is the same and the rated current (or power) is at least as high, you will be fine. It is saying that you may use 9.6A, but the supply can give you 10.83 if necessary.
If it was the other way around, the power supply couldn't put out enough juice.