I am an IPhone user and also own an Android tablet (the Lenovo A1 tablet). I was told that I should not root them. Why is this so?
"Jailbreaking" or "rooting" that means giving the owner access to system files not normally accessed.
The point about going through an authorized application store for a device is one of trust. The vendor who operates the app store, be it Apple, Google, Lenovo, Microsoft, etc. has a vested interest in making sure no malicious applications (or even just useless, dodgy ones) get into their store. Some app store operators perform checks to ensure that a minimum quality level and certain behaviors by applications are enforced.
If you are thinking about rooting or jailbreaking the device in order to bypass those all those checks,you are giving up a measure of your safety. Also, if you've found a paid-for app that's being offered for free on some unauthorized "gray market" app store, it's possible that it has been modified in some way to leak your personal information, and possibliy even install a trojan horse.
It is thus not recommended for you to do so.
Having a rooted or jailbroken phone or any handheld not just exposes your device to risks for your privacy and data but also voids the warranty. Note that a jailbroken phone will not be covered on your warranty. To clarify the security risk a bit more, it is that while you have root permission you may give root permission to applications that can turn out to be rogue or leak their permission to allow untrusted applications to gain root-like permission.
There are numerous Mobile-AV products that can, however, be installed. It is suggested that the device is restored back to the Original Manufacturer's OS/Firmware before running or installing a Mobile-AV software.
"Windows Phone 7 users who want to jailbreak their phones in a Microsoft-approved fashion can now download and install ChevronWP7 Labs for $9."
According to the article, using ChevronWP7 to unlock (jailbreak) the phone will not invalidate the warranty and Microsoft support for the phone will still be available.
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It should be noted that jailbreaking/rooting may or may not be legal in your country, so check with your country's laws for specifics.
In the 2010 review of the anticircumvention rulemakings, the U.S. Copyright Office determined that bypassing a manufacturer's protection mechanisms to allow "handsets to execute software applications" is permissible:
"(2) Computer programs that enable wireless telephone handsets to execute software applications, where circumvention is accomplished for the sole purpose of enabling interoperability of such applications, when they have been lawfully obtained, with computer programs on the telephone handset."
As already stated, however, doing so will, in all probability, invalidate any guarantee or warranty on your device.