When you multiply or divide a decimal by a power of 10 (greater than 1), the decimal point shifts to the right or left, respectively. Here’s how it works:
Multiplication by Powers of 10
- Multiplying by \(10^n\) (where \(n\) is a positive integer) shifts the decimal point to the right by \(n\) places.
- For example:
- \(3.45 \times 10^2 = 3.45 \times 100 = 345.0\) (the decimal moves 2 places to the right)
- \(0.0067 \times 10^3 = 0.0067 \times 1000 = 6.7\) (the decimal moves 3 places to the right)
- For example:
Division by Powers of 10
- Dividing by \(10^n\) (where \(n\) is a positive integer) shifts the decimal point to the left by \(n\) places.
- For example:
- \(45.6 \div 10^2 = 45.6 \div 100 = 0.456\) (the decimal moves 2 places to the left)
- \(0.789 \div 10^3 = 0.789 \div 1000 = 0.000789\) (the decimal moves 3 places to the left)
- For example:
In general:
- Multiplication by \(10^n\) increases the number's size by \(n\) decimal places.
- Division by \(10^n\) decreases the number's size by \(n\) decimal places.
This concept is fundamental in working with decimal numbers and is a useful tool in scientific notation and calculations involving measurements.
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