# Multiply and divide expressions using scientific notation and exponent properties

One application of exponent properties comes from scientific notation. Scientific notation is used to represent really large or really small numbers. An example of really large numbers would be the distance that light travels in a year in miles. An example of really small numbers would be the mass of a single hydrogen atom in grams. Doing basic operations such as multiplication and division with these numbers would normally be very comber some. However, our exponent properties make this process much simpler. First we will take a look at what scientific notation is. Scientific notation has two parts, a number between one and ten (it can be equal to one, but not ten), and that number multiplied by ten to some exponent.

Scientific Notation: $a\text{ x }10^b$ where $1\leq a<10$.

The exponent, b, is very important to how we convert between scientific notation and normal numbers, or standard notation. The exponent tells us how many times we will multiply by 10. Multiplying by 10 in affect moves the decimal point one place. So the exponent will tell us how many times the exponent moves between scientific notation and standard notation. To decide which direction to move the decimal (left or right) we simply need to remember that positive exponents mean in standard notation we have a big number (bigger than ten) and negative exponents mean in standard notation we have a small number (less than one). Keeping this in mind, we can easily make conversions between standard notation and scientific notation.

Also Read : Solve systems of equations with three variables using addition/elimination

Example 1:

Convert 14, 200 to scientific notation

Solution:

Put decimal after first nonzero number

1.42

Exponent is how many times decimal moved, 4

$\text{ x }10^4$

Positive exponent, standard notation is big and our solution is

$1.42\text{ x }10^4$

Also Read : Solve, graph and give interval notation for the solution to inequalities with absolute values

Example 2:

Convert 0.0042 to scientific notation

Solution:

Put decimal after first nonzero number

4.2

Exponent is how many times decimal moved, 3

$\text{ x }10^{-3}$

Negative exponent, standard notation is small, and our solution is

$4.2\text{ x }10^{-3}$

Example 3:

Convert $3.21\text{ x }10^5$ to standard notation

Solution:

Positive exponent means standard notation big number. Move decimal right 5 places

Our solution is 321,000.

Example 4:

Convert $7.4\text{ x }10^{-3}$ to standard notation.

Solution:

Negative exponent means standard notation is a small number. Move decimal left 3 places,

And our solution is 0.0074.

Converting between standard notation and scientific notation is important to understand how scientific notation works and what it does. Here our main interest is to be able to multiply and divide numbers in scientific notation using exponent properties. The way we do this is first do the operation with the front number (multiply or divide) then use exponent properties to simplify the 10’s. Scientific notation is the only time where it will be allowed to have negative exponents in our final solution. The negative exponent simply informs us that we are

dealing with small numbers. Consider the following examples.

Example 5:

$(2.1\text{ x }10^{-7})(3.7\text{ x }10^{5})$

Solution:

Deal with numbers and 10’s separately, Multiply numbers

(2.1)(3.7) = 7.77

$10^{-7}\text{ x }10^5=10^{-2}$

Use product rule on 10’s and add exponents

$7.77\text{ x }10^{-2}$

World View Note: Archimedes (287 BC – 212 BC), the Greek mathematician, developed a system for representing large numbers using a system very similar to scientific notation. He used his system to calculate the number of grains of sand it would take to fill the universe. His conclusion was $10^{63}$ grains of sand because he figured the universe to have a diameter of $10^{14}$ stadia or about 2 light years.

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