Beowulf cut of Grendel's head as a final act of revenge. This was done as revenge for all of the people Grendel had killed.
Beowulf cuts off Grendel's head to end the creature's reign of terror on Heorot and the people of Hrothgar's kingdom. It symbolizes Beowulf's victory over evil and his heroic triumph in vanquishing this monstrous threat.
grendel gets away, but he is fatally wounded and goes to his den to die.
Beowulf was the victor in the battle against Grendel. He ripped off Grendel's arm, causing the creature to flee back to its lair where it ultimately died.
Beowulf cut off Grendel's head to kill him and end the monster's threat to the Danes' kingdom. Grendel had been terrorizing the kingdom for years, killing and causing chaos, and Beowulf wanted to finally defeat him in battle. Cutting off Grendel's head was a way to ensure he wouldn't come back to harm anyone again.
After Beowulf cut off Grendel's arm, Grendel fled back to his lair where he eventually succumbed to his injuries and died.
Beowulf attacks Grendel, rips his arm off hangs it on the raftors and Grendel walks off and dies in his cave. Later Beowulf finds Grendel's body and cuts his head off.
Grendels arm If its after the fight with Grendel's Mother then he took Grendel's head, the sword-hilt of the sword he used to slay Grendel's mother and cut off Grendel's head, and of course Hrunting the sword he originally brought to slay Grendel's Mother which was ineffective against her tough hide.
He dies in the sea-cave lair of his mother. The open wound of his armless shoulder leads the Danes and the Geats with Beowulf to the spot, but they do not actually see the body at that time. Only after the second attack on Heorot does Beowulf find Grendel's body, at which point (presumably not longer protected by his charms as in life) Beowulf cuts off Grendel's head and returns with it to Heorot.
When Beowulf kills Grendel's mother at her lair, he sees Grendel's dead body. Out of anger, he decapitates Grendel's head and swims back to surface to show the Geats and the Danes the "trophy" he brought back.
After Grendel's mother attacked Heorot in revenge for her son's death, Beowulf pursued her to her underwater lair. There, they engage in a fierce battle, with Beowulf ultimately emerging victorious by using a giant's sword to decapitate the she-monster.
After Grendel is defeated by Beowulf, his mother seeks revenge and attacks Herot Hall. Beowulf ventures to her underwater lair and slays her as well. Beowulf returns victorious to Geatland and eventually becomes king.
Hrunting is not the name of the sword, because Hrunting bounced off of the sea-hag. Therefore, it is not the answer. The sword that killed her has no name. They had just called it, "The sword made by a giant."
He had to cut open her chest and rip out her heart.