Multiplying by a power of ten shifts the decimal point to the right (making the number larger), and dividing by a power of ten shifts it to the left (making the number smaller). The number of places shifted equals the exponent. So 3.47 x 100 = 347 (shift right 2 places) and 3.47 / 10 = 0.347 (shift left 1 place). This is a direct consequence of place value: each shift moves every digit into the next-higher (or next-lower) position. This skill is fundamental for decimal operations, metric conversions, and scientific notation.
Use a place-value chart with a fixed decimal point and slide the digits left or right to show multiplication or division by 10, 100, 1000. This visual makes it clear that the digits move, not the decimal point (though the practical shortcut is to "move the decimal point"). Practice chains: start with 4.5, multiply by 10, multiply by 10 again, then divide by 100 to return. Connect to metric conversions (meters to centimeters is x100).
You've already learned that our number system is built on place value: each position in a number is worth exactly 10 times the position to its right. The ones place is worth 1, the tens place is worth 10, the hundreds place is worth 100, and so on. Moving to the right works the same way in reverse — tenths are worth 1/10, hundredths are worth 1/100. Multiplying or dividing by a power of ten is simply the act of shifting every digit into the next-higher or next-lower position, and the number of positions shifted equals the exponent of 10 you're multiplying or dividing by.
When you multiply by 10, every digit becomes worth 10 times as much, which means each digit moves one position to the left. The 3 that was in the ones place (worth 3) now sits in the tens place (worth 30). When you multiply by 100 = 10², every digit shifts two places left. When you multiply by 1,000 = 10³, every digit shifts three places left. The digit values themselves don't change — their position does, and position is what determines value in our number system.
Dividing by a power of ten is the exact reverse: every digit shifts to the right. Dividing by 10 moves each digit one place right, making it worth one-tenth as much. So 347 ÷ 10 = 34.7: the 3 drops from hundreds to tens, the 4 drops from tens to ones, and the 7 drops from ones to tenths. The decimal point is just a marker that separates the whole-number part from the fractional part — it doesn't actually move. What moves are the digits relative to that fixed marker. The "move the decimal point" shortcut works, but understanding why it works — digits shifting positions — is what prevents errors and builds flexibility.
This skill connects directly to metric conversions, because the metric system is built on powers of ten. Converting 3.47 meters to centimeters means multiplying by 100 (there are 100 cm in 1 m): 3.47 × 100 = 347 cm. Converting 5,200 millimeters to meters means dividing by 1,000: 5,200 ÷ 1,000 = 5.2 m. Every time you see a metric prefix (kilo-, centi-, milli-), you're seeing a power of ten written as a word. Fluency with this skill makes unit conversion a matter of identifying the right power of ten and shifting — no formula required.