Question

How do decimals change when multiplied or divided by powers of 10 greater than 1

Answer 1

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)

### 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)

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.