Look at the power of 10 in the multiplier. It tells you how many zeros there are. For example, 59 * 10^2, shows that there are two zeros, making it 100.
Add two zeros to 59, making it 5,900. Now consider the problem 59 * 10^3. Add three zeros to get 59,000. In multiplying, you add as many zeros to the number being multiplied as the number of the power (the index).
Move the decimal point to the right as many times as the index indicates in performing multiplication. All numbers end in a decimal point. Generally, you would not write the decimal point unless there is a fractional component to the number. For example, 59 * 10^3 is the same as 59.000 * 10^3. Move the decimal three places to the right to obtain 59,000.
Move the decimal point to the left as many times as the index indicates in division. For example, to divide 580.0 by 10, you move the decimal point one place to the left, resulting in 58.0 as the answer. Now consider 580 divided by 10^2 (100); the answer is 5.8. In dividing by 1,000 (10^3), the answer is 0.58.
Look at the numbers to make sure you are performing the correct arithmetic operation. Another way of showing division is by writing with a negative exponent. So 10^-2 indicates division. For example, 580 * 10^-2 is the same as saying 580/10^2 (100). The answer is 5.8. The decimal point moves to the left.