Multiplying and Dividing by Powers of 10
Multiplying by 10
Multiplying a decimal by 10 makes every digit ten times bigger, which moves each digit one place to the left in the place-value columns. The simplest way to see this is that the decimal point appears to move one place to the right: 3.5 times 10 becomes 35, and 0.42 times 10 becomes 4.2. No long multiplication is needed, because place value does the work. Understanding that multiplying by 10 shifts digits, rather than adding a zero, is the foundation for multiplying and dividing by any power of 10.
Powers of 10
Ten, one hundred and one thousand are powers of 10, and each extra zero means one more place to move. Multiplying by 100 moves the point two places right, and by 1000 moves it three, because 100 has two zeros and 1000 has three. So 2.5 times 100 is 250, and 0.6 times 1000 is 600. The rule is wonderfully simple: count the zeros in the power of 10, and move the decimal point that many places. This pattern works for any power of 10, however large.
Dividing by powers of 10
Dividing by a power of 10 works the same way, but in the opposite direction: the digits become smaller, so the decimal point moves left instead of right. Dividing by 10 moves it one place left, by 100 two places, by 1000 three. So 45 divided by 10 is 4.5, and 320 divided by 100 is 3.2. When there are not enough digits, zeros fill the gap: 7 divided by 10 is 0.7. Multiplying moves the point right and dividing moves it left, by the number of zeros each time.
Multiplying in practice
Putting the rule to work, multiplying a decimal by a power of 10 is just a matter of counting zeros and shifting the point right. For 3.5 times 10, one zero means one place, giving 35; for 0.4 times 100, two zeros give 40; for 1.2 times 1000, three zeros give 1200. The key is to track the point carefully and to write in extra zeros where places run out. A multiple of a power of 10, such as 300, is a times fact and a power of 10 together, since 300 is 3 times 100. For 0.6 times 300, the fact 6 times 3 is 18 gives 0.6 times 3 = 1.8, and multiplying by 100 then moves the point two places right, giving 180. Because no calculator is needed, this is a fast, reliable method that rests on place value and the multiplication facts.
Dividing in practice
Dividing a decimal or whole number by a power of 10 follows the mirror-image rule: count the zeros and shift the point that many places left. So 45 divided by 10 is 4.5; 600 divided by 100 is 6; and 8 divided by 10 is 0.8, with a zero written in front to hold the ones place. Watching where the point lands, and adding leading zeros when needed, keeps the value correct. To divide by a multiple such as 20, divide by 2 and then by 10: 4.8 divided by 2 is 2.4, and 2.4 divided by 10 is 0.24. Dividing this way is exactly as quick as multiplying, once the direction of the shift is clear.
Estimating to check
Because it is easy to move the point the wrong number of places, estimating is a vital check. A quick estimate, using rounding and place value, tells you the size the answer should be: 4.9 times 10 should be about 50, so an answer of 49 is reasonable while 490 would be far too big. A common error is treating multiplying by 100 as simply adding two zeros, which fails for decimals; place-value reasoning shows the point must move two places. Checking each answer against an estimate catches these slips before they cause trouble.
Where this leads
Multiplying and dividing by powers of 10 is the engine behind the metric system, scientific notation, and everyday work with money and measurement, where converting between units means shifting the decimal point. The place-value thinking built here, that each power of 10 moves the point a fixed number of places, carries directly into multiplying and dividing decimals by any number, and into the powers and indices of later mathematics. It is a small rule with very wide reach, used constantly in science, finance and daily life.