2.11 The laws of exponents and scientific notation
Introduction
We've previously looked at how to use scientific notation to write really big or really small numbers. Remember they are written in the form a\times 10^{b}, where 1 \leq a < 10 and b is an integer.
Since these numbers are all written in relation to a power of 10, we can simplify expressions written in scientific notation using the laws of exponents, such as the product of powers property or the division of powers property .
Ideas
Operations with numbers in scientific notation
If we are multiplying or dividing numbers written in scientific notation, we will use the commutative property to multiply or divide the powers of 10 by one another and then the "a and b" values by one another.
| \displaystyle (a \times 10^{m})( {b \times 10^{n}}) | \displaystyle = | \displaystyle (a \times b)(10^{m} \times 10^{n}) |
| \displaystyle = | \displaystyle (a \times b) \times 10^{m+n} |
| \displaystyle (a \times 10^{m})\div( {b \times 10^{n}}) | \displaystyle = | \displaystyle (a \div b)\times(10^{m} \div 10^{n}) |
| \displaystyle = | \displaystyle (a \div b) \times 10^{m-n} |
If you are adding or subtracting numbers written in scientific notation, you need to make sure that the powers of ten are the same. To do this, you may need to factor some of the powers of 10 based on the product of powers property. Then you can use the decimal number, rather than the number in scientific notation and evaluate the problem. The process goes:
| \displaystyle (a \times 10^{n}) \pm (b \times 10^{m}) | \displaystyle = | \displaystyle (a \times 10^{n}) \pm (b \times 10^{m-n} \times 10^{n}) | Write both numbers as multiples of 10^n |
| \displaystyle = | \displaystyle (a \pm b \times 10^{m-n}) \times 10^{n} | Factor out 10^n |
Now we can perform the calculation inside the parentheses.
Let's look at the following examples in evaluating operations with scientific notation.
Examples
Example 1
Use the laws of exponents to simplify (2 \times 10^{6}) \times (6 \times 10^{5}). Give your answer in scientific notation.
Example 2
Evaluate 1.77 \times 10^{9} + 3.24 \times 10^{10}.
Give your answer in scientific notation and in three decimal places.
Example 3
Use law of exponents to simplify \dfrac{3\times 10^{3}}{12 \times 10^{-1}}. Give your answer in scientific notation.
To add or subtract numbers in scientific notation, factor out a power of 10.
To multiply or divide numbers in scientific notation, use the commutative property to multiply or divide the powers of 10 by one another and then the decimals by one another.
Powers of numbers in scientific notation
Recall the power of a power and power of a product properties. We will use both properties to raise a number in scientific notation to a power. The process is as follows:
| \displaystyle (a \times 10^{m})^{n} | \displaystyle = | \displaystyle a^{n} \times (10^{m})^{n} |
| \displaystyle = | \displaystyle a^{n} \times 10^{mn} |
After completing the above process, make sure to check that your final answer is expressed appropriately in scientific notation and simplify as required.
Examples
Example 4
Write \left( 4 \times 10^{6}\right)^3 in scientific notation.
We can use the power of a power and power of a product properties to raise a number in scientific notation to a power:
| \displaystyle (a \times 10^{m})^{n} | \displaystyle = | \displaystyle a^{n} \times (10^{m})^{n} |
| \displaystyle = | \displaystyle a^{n} \times 10^{mn} |
Appropriate units of measurement
Choosing appropriate units is important in many situations, from simple measurements in cooking to advanced measurements in astronomy, physics, and chemistry.
Examples
Example 5
Neil is looking at a table of distances between the sun and heavenly bodies. He is interested in the distance between the sun and the nearest star (Proxima Centauri) which is represented by the number 4.013 \times 10^{13}.
Which of the following could be the appropriate unit for the given distance between the sun and Proxima Centauri?
Using scientific notation and choosing appropriate units are important skills to report measurements properly.