Scientists frequently deal with very large and very small numbers and so will you. It is easier to use exponential notation than to work with the large and small numbers involved. 1,000,000,000 equals 10⁹, but 10⁹ is easier to use.

An exponent shows the number of times the base is to be multiplied by itself. For example 4³ = 4 x 4 x 4 = 64. In this example, 4 is the base and 3 is the exponent Most often, in your science numbers, the base used is 10. For example, 10² = 10 x 10 = 100

A negative exponent represents the reciprocal of the number obtained when the base is raised to the positive value of the exponent. For example:
10^-2 = = 0.01
For any base, if the exponent is zero, the quantity is 1. For example 10⁰ = 1

When exponential expressions are multiplied, the exponents are added: (a^m)(aⁿ) = a^(m+n)

For example: (10²) (10³) = 10⁽²⁺³⁾ = 10⁵

When exponential expressions are divided the exponents are subtracted: (a^m)/(aⁿ) = a^(m-n)

For example: (10²) / (10³) = 10^(2-3) = 10^-1

This helps explain how negative exponents work. In the previous example

10^-1 = (10²) / (10³) = 100 / 1000 = 1 / 10 = 0.1

And similarly, 10⁰ = 1 because, for example , 10³ / 10³ = 10^(3-3) = 10⁰ = 1

When an exponential expression is raised to a power, the exponent is multiplied by the power:

(a^m)ⁿ = a^mn

Scientific Notation

The form in which exponential quantities are most often expressed in science is known as scientific notation. A number written in scientific notation is in the form a x 10^b . Where a is equal to or greater than 1 and less than 10 and b is either a positive or negative integer.

To write the speed of light in a vacuum, 300,000,000 m/s in scientific notation, a = 3 and b = 8 (the number of places you move the decimal point to the left). Giving you: 3 x 10⁸ m/s.

The diameter of a human hair is about .00011 m. In scientific notation, a = 1.1 and b = -4 (the number of places you move the decimal point to the right). Giving you: 1.1 x 10^-4 m

Notice that the a part of the notation contains the same number of significant digits as the original measurement.

When changing a number back from scientific notation remember that if b is positive, the original number is greater than 1, and if b is negative, the original number is less than 1. This indicates which direction to move the decimal point.

It is not necessary to un-scientific notation a measurement. Scientific notation is a valid number and scientific calculators can handle this notation. If your calculator does not have an exponent function do this: do the calculation for the a of each measurement and then the 10b of each measurement. Then adjust your final answer to be in the proper scientific notation format. (1.2 x 10⁴ m) (9.4 x 10³ m) 1.2 x 9.4 = 11.28, (10⁴) (10³) = 10⁽⁴⁺³⁾ = 10⁷ together you have 11.28 x 10⁷ m².

But alas, this is not in the proper format, a must be less than 10. So….. 1.128 x 10⁸ m². What about significant digits?
Okay, 1.1 x 10⁸ m²