Otto’s pretty well correct – the laws governing Powers of Attorney in your state and the specific wording of your Power of Attorney make all the difference in the world.
In general, a Power of Attorney is a document granting sweeping powers to one person to act in behalf of another, executed by the second when in full possession of mental powers to enable the first to act at a time and place when he/she cannot. It’s an extension of the idea of agency – if I, wishing to do something at two separate places effectively simultaneously, hire you to do one of those things, you are acting as my agent in doing that thing.
Powers of Attorney can be specific or general. When Robert A. Heinlein went in for brain surgery, he provided two powers of attorney – a general one for his wife, and one specifically limited to his literary/business affairs to his longtime agent, enabling either to make publishing decisions and her to make life-and-home decisions in his behalf and as though they were he.
Regarding the rent, though, one quick item of advice: There are a whole string of clauses, probably, stating what you can do in your mother’s behalf. If one deals specifically with her home and personal affairs, or authorizes you to pay her just debts and obligations, then you can write out a rent check on her account, print her name on the signature line, and then sign your own name below it followed by “POA.” You will probably have to provide the bank with a copy of your Power of Attorney for them to honor the check signed in that way, but that’s standard practice in such cases; an officer or teller will be familiar with procedures and can walk you through their local policies.
You should, however, go to the attorney who prepared the POA and ask him to go over the document with you and explain in plain English what you are authorized to do. Odds are your powers are far more than you would ever have expected. This is not a run-out-this-minute urgency, but you should be familiar with what you can and cannot do, and the sooner the better.
Saying she’s “out of it” raises the question of whether she may require being placed in a nursing home (in the absence of family who can provide 24-hour care) – and it’s my impression that state laws differ greatly on what it takes for you to make that decision. Your POA may permit it, or you may have to go before a court with her doctor as expert witness to get them to authorize it (unless she is of clear enough mind to consent and does consent to the placement – and be sure to have witnesses to that fact!).
There is one other item about POAs, which carries a veiled and unintended insult with it. You are legally required to act in her best interests when exercising it. (The veiled insult is in the implication that you otherwise might not.) I bring this up because there is a very strong distinction – you may make “customary” discretionary payments, such as a church pledge or membership dues to something she’s belonged to for a long time, but you may not give one cent of her money to a charity or to family, even if you know it would be her wish to do so. Your job is to conserve her assets and act solely in her best interests.