Use sharp scissors to trim brown leaves. Over or under watering can cause yellowed or brown leaf tips. These can be snipped off with a pair of sharp scissors or small pruning shears. Cut the leaves on a slight angle to prevent splitting and to keep the leaves' shape.
Use a handsaw to cut off a dead head. Many ponytail owners keep their plants outside throughout the spring and summer, and if a palm is left out too long, an early frost can severely damage the "head" or leaf crown. If this happens, you may need to saw off the entire crown. Use a sharp handsaw or reciprocating saw to trim the head off as close to the damaged area as possible--your goal should be to conserve as much of the trunk as possible.
Seal the cut to protect it. If you prune off the entire crown, use a tree pruning sealant or candle wax to thinly coat the cut. This will facilitate healing and also protect the fresh cut from insect damage and excessive moisture-loss.