Underwater, in a lake.
Grendel's mother kills Hrothgar's best friend.
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In the EPIC of Beowulf, she was slain by a sword made for a giant and sliced her head clean off in one clean cut.
Her son's, Grendel, arm back that they took as a "victory" of killing him, and the sword that they used on him .
A swamp
Hrunting, a sword handed down from Hrothgar's forefathers.
After Beowulf kills Grendel's mother, he sees Grendel's lifeless body and beheads it as proof of his victory. He then finds the decapitated head of Grendel and takes it with him as he returns to Heorot to show it to Hrothgar. The Geats and Danes rejoice in Beowulf's triumph over the monsters.
While Grendel's mother is hard for Beowulf to defeat, he IS able to use a sword, where as he has to battle Grendel bare handed.
he dies later on
Under water, where Grendel's mother lived.
Beowulf kills Grendel's mother by using her own weapon, a giant's sword, against her. This is ironic because he defeats her with the very weapon she was planning to use against him. Additionally, he avenges Grendel's death by killing his mother in the same manner that Grendel killed Hrothgar's men.