The Idea in Brief
Jewish oral tradition as documented in the Talmud indicates that the "outstretched arm and hand" of Pharaoh's daughter (that saved Moses) was the same imagery of the "outstretched arm and hand" (that had saved Israel and punished Egypt). That is, Pharaoh's daughter had rejected the idolatry of Egypt, and when she approached the Nile to wash, she had disobeyed the edict of her father to save the Hebrews, since the handmaidens had recognized the basket as containing a Hebrew child. That is, Pharaoh's daughter proceeded to save Moses with an "outstretched arm and hand," which was the larger metaphor for HaShem, who saved the Israelites with an "outstretched arm and hand."
The following captures Rashi's comments, and provides additional information as found in the Talmud and Dead Sea Scrolls.
To start, the Babylonian Talmud indicates that Pharaoh's daughter had rejected the idolatry of Egypt. That is, the Gemarah of the Talmud (b. Sotah 1:8-9, III.32 [Folio 12B]) reports that oral tradition indicates that her washing in the Nile was an allusion to her inner cleansing from the idolatry of Egypt.
"And the daughter of Paraoh came down to bathe at the river" (Ex. 2:5):
Said R. Yohanan in the name of R. Simeon b. Yohai, "This teaches that she came down to wash off her body the filth of her father's idols.
"And so Scripture says, 'When the Lord will have washed away the filth of the daughters of Zion' (Is. 4:4)."
The same passage in the Talmud indicates that the handmaidens, on the other hand, who accompanies her, had walked in darkness.
"And her serving girls walked along" (Ex. 2:5):
Said R. Yohanan, "This 'walking' refers only to death,
and so Scripture says, 'Lo, I am walking towards death (Gen. 25:32)."
These handmaidens alluded to the edict of the Pharaoh; that is, not even the daughter of Pharaoh (royal family member) could disobey the edict to leave Hebrew children for dead (by exposure to the elements). We read the following according to b. Sotah 1:8-9, III.32 [Folio 12B].
"And she say the ark among the reeds" (Ex. 2:5):
When [her companions] saw that she wanted to save Moses, they said to her, "Our mistress, it is customary in the world that, when a mortal king makes a decress, even though everyone in the world may not carry it out, his sons and the members of his household carry it out.
"But you are going to violate the decree of your father!"
Gabriel came along and beat them into the ground.
The angel Gabriel thereupon destroyed them, since they would have betrayed Pharaoh's daughter for disobeying the edict of Pharaoh.
Next, this passage in the Talmud mentions the lengthening of the arm, but the careful reading of this passage reveals something unusual. The rabbis allude to Psalm 3:8, where the arm is outstretched to strike the face of the enemy.
"She sent her serving girl to get it" (Is. 2:5):
R. Judah and R. Nehemiah:
One said, "She sent forth her hand."
The other said, "She sent forth her serving girl."
The one who said, "She sent forth her hand," [cites] that which is written [in the verse at hand], "Her arm."
The one who said, "She sent forth her serving girl" points out that the world hand is not written.
But in the view of him who said that it was her serving girl, has it just now not been stated that Gabriel came and beat them into the ground? [So where would her serving girl come from?]
[The answer is] that he left her once, since it is not appropriate that a princess should stand all by herself [unattended].
And according to the one who said, "Her hand," should not Scipture have written, "Her hand"?
In this way we are informed that her arm was lengthened.
For a master has said, "And so you find in the case of the arm of the daughter of Pharaoh, and so you find in the case of the teeth of the wicked.
"For it is writte, 'You have broken the teeth of the wicked' (Ps. 3:8).
"In this connection, R. Simeon b. Laqish said, "Do not read, 'you have broken, but, 'you have lengthened.'"
The teeth in Psalm 3:8 allude to might and power according to Rashi.
In other words, the Talmudic teaching is that the "outstretched arm and hand" is both an agent of judgment (reaching out and striking the enemies of HaShem) as well as an agent of deliverance (reaching out and saving the people of HaShem). The Talmudic rabbis therefore understood Jewish oral tradition to refer not to the handmaidens of Pharaoh's daughter who intercepted Moses in the Nile River, but to Pharaoh's daughter herself. The image below comes from the Dead Sea Scrolls. This particular miniscule (4Q13Exodusb) is the oldest surviving portion of שְׁמוֹת (Exodus) in Hebrew, which was written sometime around the first century BCE.
Please click the image to enlarge.
Special infrared light enables better examination of the same miniscule.
Please click the image to enlarge.
When we zoom in on the words in question, we see the following -- that is, there are no dagesh marks or any other indicators to help understand whether we are talking about handmaidens or outstretched arms -- in other words, Jewish oral tradition pointed toward the "outstretched arm and hand" of Pharaoh's daughter, and not to her handmaidens that pulled Moses out of the water. The ambiguity existed in the absence of vowel and consonant pointing.
Please click the image to enlarge.
Please note that the same Hebrew verb in this verse וַתִּשְׁלַח occurs seven times in Torah with the meaning of "stretching out the hand." Finally, in the Masoretic Text, there is no Sof Pasuq cantillation at the end of this verse! (The Sof Pasuq looks like the English colon [:] mark.) There is no Sof Pasuq to indicate the logical end of the verse in the Masoretic Text!
In other words, the Masoretic editors (sometime before or during the tenth century CE) interpreted the reading of Pharaoh's daughter sending her handmaidens into the water, and so they scripted the verse cantillation markings accordingly. However, by omitting the Sof Pasuq, they allow ambiguity of oral tradition. How do we know this!?
When we return to the Dead Sea Scrolls, we note that in the following verse, there is a marked difference from the Masoretic Text.
The Masoretic Text says
וַתִּרְאֵהוּ אֶת הַיֶּלֶד (she saw him the lad)
But the miniscule from Dead Sea Scrolls says
ותראה את הילד (she saw [the presence of the Holy One] upon the lad)
This anomaly finds resolution in the Talmud as follows (Sotah 1:8-9, III.36 [Folio 12B]), which would reflect the reading as found in the image of the Dead Sea Scrolls (above):
"She opened it and saw him, the child" (Ex. 2:6):
The text should have said simply, "She saw [the child], [omitting "him"].
Said R. Yosé b. R. Hanina, "She saw the Presence of God with him."
The Talmud existed before the vowel marks and cantillation of the Masoretic Text by several hundred years. Therefore, there is no surprise to see the Masoretic editors omit the Sof Pasuq (from the preceding verse) in order to allow the interpretation and reading from the Talmud!
The lesson from this portion of Torah is that the "outstretched arm and hand" of Pharaoh's daughter (that saved Moses) was the same imagery of the "outstretched arm and hand" (that had saved Israel and punished Egypt). Thus as noted in the Talmud reference to Psalm Psalm 3:8 the "outstretched arm and hand" breaks the power ("teeth") of the enemy. So the when the Red Sea collapsed upon the Egyptian Army, the same power delivered the Israelites from the enemy of HaShem. In this light, Pharaoh's daughter saw not only the child Moses, but the very presence of HaShem with him, who would lead the Israelites in disobedience to Pharoah.
The current Masoretic Text, however, is "not incorrect" in any grammatical sense (considering that when adding vowel and cantillation points, they had to determine set meanings). In this respect, the Masoretes left several "markers" in the Masoretic Text to help readers and teachers look into the Talmud for the oral tradition. Thus they omitted the Sof Pasuq.
In conclusion, the Hebrew language did not have vowel and cantillation points in antiquity as evidenced by the Dead Sea Scrolls, which preceded the Masoretes by almost 1,000 years. However, Jewish oral tradition, as contained in the Talmud, provides the background and texture on how to approach and understand the Scriptures in order to develop its fullest and freshest meanings, which are relevant for us all today.