Numbers with multiple zeros can be difficult to record and manipulate. Consequently, scientists and mathematicians use a shorter method to write significantly large or small numbers called scientific notation. Instead of saying the speed of light is 300,000,000 meters per second, scientists can record it as 3.0 x 10^8. Simplifying the numbers makes them not only easier to express, but also easier to multiply.
Using Scientific Notation
To write a number in scientific notation, you must write it as the product of a number and a power of 10. The first number is called the coefficient, and it must be greater than or equal to 1 and less than 10. The second number is called the base, and it is always written in exponent form. To convert a number to scientific notation, put a decimal after the first digit. This becomes the coefficient. Then, count the number of places from the decimal to the end of the number. This number becomes the exponent. For the number 987,000,000,000, the coefficient is 9.87. There are 11 places after the decimal, so the exponent is 11. In scientific notation, it is 9.87 x 10^11.
To multiply numbers in scientific notation, first multiply the coefficients. Then, add the exponents of the two numbers and keep the base 10 the same. For example (2 x 10^6) (4 x 10^8) = 8 x 10^14.
Adjusting the Coefficient
Remember, the coefficient must always be a number between 1 and 10. If you multiply the coefficients and the answer is greater than 10, you must move the decimal and adjust the exponents accordingly. When you multiply (6 x 10^8)(9 x 10^4) you get 54 x 10^12. Move the decimal, so the coefficient becomes 5.4 and add one exponent to the power of 10. The final answer is 5.4 x 10^13.
Scientific notation is also used to write very small numbers. For these numbers, the format is the same, but negative exponents are used. The number 0.00000000001 is written as 1.0 x 10^-11. The -11 signifies that the decimal point is moved 11 places to the left of "1."
Multiplying With Negative Exponents
To multiply numbers in scientific notation when the exponents are negative, follow the same rules as simple multiplication. First, multiply the coefficients and then add the exponents. When adding the exponents, use the rules of addition for negative numbers. For example, (3 x 10^-4) (3 x 10-3) = 9.0 x 10-7. When one exponent is positive and one is negative, subtract the negative from the positive number. For example, (2 x 10^-7)(3 x 10^11) = 6.0 x 10^4.