Final answer:
Dan incorrectly expanded the number two million, three hundred fifty thousand, four, due to a misunderstanding of powers of ten. The correct expanded form is (2x10⁶)+(3x10⁵)+(5x10⁴)+4, where each component represents a different place value correctly based on the number of zeros.
Step-by-step explanation:
The mistake made by Dan in writing the expanded form of two million, three hundred fifty thousand, four is in misunderstanding the powers of ten. Dan wrote (2x10⁶)+(3x10⁴)+(5x10³)+4 whereas the correct expanded form should be (2x10⁶)+(3x10⁵)+(5x10⁴)+4.
He confused the power of 10´ as 10,000 instead of 1,000 and represented three hundred thousand as 3x10⁴ instead of 3x10⁵. Similarly, fifty thousand was mistaken for 5x10³ instead of the correct 5x10⁴. Understanding that the exponent indicates the number of times 10 is multiplied by itself is crucial, and in this case, Dan should have realized that for fifty thousand, 10 needs to be multiplied four times (10x10x10x10), not three.
To avoid such errors, remember that each digit in a number refers to a power of ten based on its position. The number immediately before any zeros tells you how many sets of that power of ten you have. So, 2 million is 2 sets of a million (10⁶), 300,000 is 3 sets of one hundred thousand (10⁵), 50,000 is 5 sets of ten thousand (10⁴), and so on.