Press the 2nd Key (first column, second row on the TI-89's keypad), then press the "On" key (first column, 10th row) to turn your calculator off.
Slide the TI-89's back cover off to reveal the back side of the calculator. Locate the battery compartment cover and its associated latch, directly above the battery cover.
Insert the tip of your index finger in the battery latch cavity. Press the tip of your finger firmly down so that it moves in a direction parallel to the back plane of the calculator.
Keep depressing your index finger tip against the latch until it has no more give. Pull the latch with the tip of your index finger in a direction that is nearly perpendicular to the back of the calculator case.
Remove the battery compartment cover as you are in the process of prying it off with your index finger. Place the battery cover on the table.
Locate the enclosure for the backup battery. Look slighty above the main battery bank of 4 AAA batteries on the left side. Note that the battery backup enclosure cover reads "Back up battery."
Unscrew the screw that secures the battery backup cover with a small Philips screwdriver. Place the screw and cover on top of the main battery cover you placed on the table,
Remove the backup battery with your fingers. Insert a new backup battery (with part number SR44SW or 303) into the backup battery cavity. Ensure that the backup battery's positive side, labeled with a "+" sign, is still visible (face up) when it is inserted into the battery cavity.
Screw the backup battery cover back on. Slide the main battery cover back on. Press down on the battery cover to snap it into place. Press the "On" key (column 1, row 10). Verify that your display lights and your calculator work. Adjust the screen display lighting if the screen cursor cannot be seen or cannot be seen clearly.