Can I root without unlocking bootloader?

No, roots needs bootloader unlock but if you want to don’t flash custom rom then you can use stock rom ya xiaomi.eu rom. 

What is xiaomi.eu rom?
xiaomi eu is clone of xiaomi MIUI rom form others region. it is a custom rom which gives you advance and without bloatware. (not given by xiaomi but developers from community) 
xiaomi eu rom is based on china rom.
How to unlock bootloader?
In order to install custom rom and root your phone you need to unlock bootloader.
xiaomi user unlock bootloader through mi unlock tools. mi unlock tools gives by xiaomi. you need wait about 7 days after getting permition to unlock bootloader.
Step 1 Make a xiaomi account and sync your moblie number.
xiaomi user generally skip making mi account during the initial setup. but in  order to unlock the Mi device bootloader, it is necessary to have a working account on the device.
Moreover, it is important to link the number on the device with the xiaomi account, Also, refrain from connecting unowned numbers to xiaomi account on the deevice in order to avoid obstacles in process.

After falling to unlock the bootloader a few times, realized that Xaiomi requires ” Find My Device” toggled and synced to the latest location for some reason.   
Can I root without unlocking bootloader? - Stylemecck
You will Find the setting under mi account> mi cloud>Find device. In order to update location, sign in at Xiaomi cloud website on a computer, click on “find Device” and choose your current device.
Step 2 – Authorize Mi Unlock in the developer settings.
In tis step, you have to bind your account with the Mi Unlock authorization tab which you can  find under the developer settings.
Open the device settings > Tap on About phone > Scroll down and tap, on the MIUI verson slot unlit ” devloper” doesn’t actives about tap 5 times. This will unlock the developer  options in your MIUI device which can be located under “Additional Settings.”
Can I root without unlocking bootloader? - Stylemecck

In the Developer option, go to the “MI unlock status” and tap on “Add account and device ” in order yo authorize  Xiaomi to unlock the device associated with the account.

Can I root without unlocking bootloader? - Stylemecck

The Authorization process will use the mobile data of the carrier.Therefore , make sure you have turned off WiFi before addding the account.

Before moving ahead with the third step, enable OEM unlocking and USB debugging while your are in the third step are in the developer option settings.
Step 3 – Download the MI Unlock 

Keep Your xiaomi or Redmi phone aside for a moment and go to this link on your pc to download your pc to the Mi unlock Tool.
Extract the zip file and open the application named ” Mi Unlock Flash File”. Log in using the same Mi account which is being used on the Xiaomi smartphone.
Can I root without unlocking bootloader? - Stylemecck

Before starting with the Mi bootloader unlocking process, tap on the top-right gear icon to check the update for both the USB driver and MI Unlock Tool.

Can I root without unlocking bootloader? - Stylemecck

As instruction on the app screen, switch off your smartphone. Take a moment and hold the power button and lower volume button simultaneously. Thid will open up the fastboot mode in your MIUI device. Now, Connect the phone to the computer via USB cable.

Can I root without unlocking bootloader? - Stylemecck

A new message will appear on the screen, “Phone connected”. Now, click on the “unlock” button to unlock the bootloader.

Can I root without unlocking bootloader? - Stylemecck

Step 4 – Wating for Xiaomi to end the unlock period

ON your first attempt, the Mi unlock tool will flash the message, “ouldn’t unlock, Please unlock 168 hours later”. For thoes Who don’t know, this is the infamous Xiaomi bootloader unlocking wating period where, sometimes, users end up wating for months
Can I root without unlocking bootloader? - Stylemecck
However, several users claim to get the unlock device authorization merely within a few hours. If you want to avoid the same fate, just try not to bypass the 168 hours error.
The Mi Unlock tool waiting period is a bitter pill to swallow. There is no workaround, in fact, you might wind up resetting the clock trying to trick the system.
The best thing you can do is wait patiently and once the unlock period is over, again use the Mi unlock tool and hopefully, unlock the bootloader of your MIUI device.
Can I root without unlocking bootloader? - Stylemecck

Step 5  – Check the Mi unlock staus on your device

Can I root without unlocking bootloader? - Stylemecck

By now, I am guessing you have successfully unlocked the bootloader of your Redmi device. Before you head on to do other things such as flashing the MIUI device or rooting the device, do a quick check if the unlock procedure was a success.

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