You have seen that a power of 10 like $10^{2}$ is $100$. Multiplying or dividing by 10, 100 or 1,000 is quick, because of how place value works.
Idea clave
Multiplying by 10 moves every digit one place to the left; dividing by 10 moves every digit one place to the right. For 100 or $10^{2}$, move two places; for 1,000 or $10^{3}$, three. Fill any empty places between the digits and the point with zeros.
Por qué funciona
Multiplying by 10 makes each digit worth 10 times as much, so each digit moves into the place that is worth 10 times as much: the place to its left. The decimal point stays put; the digits move. It looks as if the point moved the other way, and you can think of it that way too.
Worked examples
Ejemplo resuelto: Multiply a decimal by 100
Find $3.6 \times 100$.
- 100 is $10^{2}$, so each digit moves two places left.
- The 3 (ones) becomes 3 hundreds; the 6 (tenths) becomes 6 tens.
- The ones place is now empty, so write 0.
Respuesta: $360$
Ejemplo resuelto: Divide by 1,000
Find $47 \div 1{,}000$.
- Each digit moves three places right.
- The 4 (tens) becomes 4 hundredths; the 7 (ones) becomes 7 thousandths.
- The tenths place is empty: write 0.
Respuesta: $0.047$
Ejemplo resuelto: Using an exponent
Find $0.52 \times 10^{3}$.
- $10^{3} = 1{,}000$: move three places left.
- $0.52 \to 5.2 \to 52 \to 520$.
Respuesta: $520$
Error común
“Adding a zero” to a decimal, like writing $2.5 \times 10 = 2.50$. A zero at the end of a decimal changes nothing: $2.50$ is still two and a half. Check: multiplying by 10 must give a bigger number. $2.5 \times 10 = 25$.