Shmos 2:5 – Pharaoh’s daughter went down to bathe by the river and her maidens walked along the river. She saw the basket among the reeds and she sent her maidservant and she took it.
Gemora Sotah 12b: And sent her handmaid to fetch it — R. Judah and R. Nehemiah [differ in their interpretation]; one said that the word means ‘her hand’ and the other said that it means ‘her handmaid’. He who said that it means ‘her hand’ did so because it is written ammathah; he who said that it means ‘her handmaid’ did so because the text does not explicitly say yadah [her hand]. But according to him who said that it means ‘her handmaid’,it has just been stated that Gabriel came and beat them to the ground! — He left her one, because it is not customary for a king’s daughter to be unattended. But according to him who said that it means ‘her hand’, the text should have been yadah! — It teaches us that [her arm] became lengthened;
Torah Temimah Colloquial Translation on Note #27:
Because she stretched out her hand in order to reach the basket. According to the one who believes that the word “ammathah” means maidservant, it is clearly because it did not write “yadah” explicitly as is often done. Further the one who thinks the word means maidservant, does not accept the teaching that her arm [hand] miraculously lengthened. The one who holds that the word means ‘hand’ it is from the phrase אמת יד which means arm. Apparently, it is difficult to understand what forced this disagreement since the simple meaning of the word ammathah is indeed maidservant.
Perhaps the Midrash is being careful in explaining the words of this posuk. In the first part of the posuk it uses the word נערתיה to mean “her maidservants”. So too in the second part of the posuk it should have used the same word to refer to the maidservants. Truly, it is the customary way to refer to a king or queens servants as נער out of respect for the king or queen instead of the using the word slave or maidservant. This is customary in the world at large and also in the Tanach. The word נער [youth or lad] is more polite than saying slave.
So from the fact that the posuk changed its phraseology from נער to אמתה (ammathah) that is the source of the explanation that it actually means that it was her hand that she “sent” and stretched out because the basket was far away from her.
Additionally, in Midrash Rabba it mentions that the Rabbis say that Pharaoh’s daughter had tzaraat (a type of leprosy). That was the reason why she went down to the river to wash. As soon as she touched the basket, she was healed. That is why she had mercy on him and loved him so much. <end quote of the Midrash>
It is not clear where is the hint in this posuk to what the Midrash says. Perhaps it is from the idea that it just says she went down to the river to wash but it doesn’t mention whether she’s going to wash her face, her hands or her feet. You’d have to say, as we do in many places in the Tanach, that she was intending to wash her whole self. In other words, her intent was to immerse in the Nile and to wash off all the tumah/impurity. Since it wouldn’t make sense to say that she was immersing for religious reasons, one is forced to say that it was a physical washing of her whole body due to the leprosy.
DBS Note: I once heard a wonderful rabbi explain that Pharaoh’s daughter knew the basket was too far for her to reach. She stretched out her hand anyway because she had to do everything she could to rescue Moshe whether it seemed rational or not. Similarly, we must strive to accomplish great things whether we think they are rationally attainable or not.