Powers of 10: Shifting the Decimal

Place values continually decrease from left to right by powers of 10. Understanding this can help you understand the following shortcuts for multiplication and division.

When you multiply any number by a positive power of 10, move the decimal forward (right) the specified number of places. This makes positive numbers larger:

3.9742 × 103 = 3,974.2	Move the decimal forward three spaces.
89.507 × 10 = 895.07	Move the decimal forward one space.

When you divide any number by a positive power of 10, move the decimal backward (left) the specified number of places. This makes positive numbers smaller:

4,169.2 ÷ 102 = 41.692	Move the decimal backward two spaces.
83.708 ÷ 10 = 8.3708	Move the decimal backward one space.

Note that if you need to add zeroes in order to shift a decimal, you should do so:

2.57 × 106 = 2,570,000	Add four zeroes at the end.
14.29 ÷ 105 = 0.0001429	Add three zeroes at the beginning.

Finally, note that negative powers of 10 reverse the regular process:

6,782.01 × 10−3 = 6.78201

53.0447 ÷ 10−2 = 5,304.47

You can think about these processes as trading decimal places for powers of 10. Think about why this is. The expression 10⁻³ is equal to 0.001. As a concrete example, if you multiply 6,782.01 by 0.001, you get a much smaller number.

For instance, all of the following numbers equal 110,700.

110.7           ×       10³

11.07           ×       10⁴

1.107           ×       10⁵

0.1107           ×       10⁶

0.01107           ×       10⁷

The first number gets smaller by a factor of 10 as you move the decimal one place to the left, but the second number gets bigger by a factor of 10 to compensate.

Check Your Skills

3. 0.0652 × 10⁻² = ?

4. 

5. Put these numbers in order from least to greatest:

   1. 234 × 10⁻²
   2. 0.234 × 10²
   3. 2.34 × 10⁴