About Operations with Scientific Notations
What is the Scientific Notation of Numbers?
Scientific notation is a standardized way to represent very large or very small numbers as the product of a number and a power of ten.
In scientific notation, numbers are written as \(a\times 10^b\), where 1 ≤ a < 10 and b is the exponent.
Adding and Subtracting with Scientific Notation
Before we can add or subtract, we must convert all powers of ten to the same value.
Example: \(5.3\times 10^6+11.2\times 10^7\)
Step 1: Rewrite the numbers so that they all have the same power of ten.
\(5.3\times 10^6=\frac{5.3\times 10}{10}\times 10^6=0.53\times 10^6\times 10^1=0.53\times 10^7\)
Hence, \(5.3\times 10^6+11.2\times 10^7\) = \(0.53\times 10^7+11.2\times 10^7\)
Step 2: Add the numbers.
\((0.53+11.2)\times 10^7=11.73\times {10^7}\)
Step 3: Rewrite the sentence in a scientific notation.
\(11.73\times 10^7=1.173\times 10^8\)
Example 1:
Calculate the sum: \((5.2\times 10^3)+(7.72\times 10^3)\)
\((5.2+7.72)\times 10^3\) The distributive property
\(=12.92\times 10^3\) Add
\(=(1.292\times 10^1)\times 10^3\) Write 12.92 in scientific notation.
\(=1.292\times 10^4\) The product of powers property
Example 2:
Calculate the difference: \((4.2\times 10^{-2})-(3.3\times 10^{-3})\).
Solution:
Rewrite \(3.3\times 10^{-3}\) to have the same power of ten as \(4.2\times 10^{-2}\)
\(3.3\times 10^{-3}=3.3\times 10^{-1}\times 10^{-2}\) Rewrite \(10^{-3}\) as \(10^{-1}\times 10^{-2}\)
\(=0.33\times 10^{-2}\) Rewrite \(3.3\times 10^{-1}\) as 0.33
Subtract the factors.
\((4.2\times 10^{-2})-(0.33\times 10^{-2})\)
\(=(4.2-0.33)\times 10^{-2}\) The distributive property
\(=3.87\times 10^{-2}\) Subtract
Multiplying in Scientific Notation
Multiply the factors and powers of ten separately to multiply the numbers written in scientific notation.
The Product of Powers Property
According to the product of power property, when multiplying two powers with the same base, we add the exponents.
\(a^m\times a^n=a^{m+n}\)
Example 3:
Find \((4\times 10^{-4})\times(6\times 10^{-3})\)
Solution:
By the commutative property of multiplication
\((4\times 10^{-4})\times (6\times 10^{-3})=4\times 6\times 10^{-4}\times 10^{-3}\)
\(=(4\times 6)\times (10^{-4}\times 10^{-3})\) The associative property of multiplication
\(=24\times 10^{-7}\) Simplify.
\(=(2.4\times 10^{1})\times 10^{-7}\) Write 24 in scientific notation.
\(=2.4\times 10^{-6}\) The product of powers property
Dividing in scientific notation
Divide the factors and powers of ten separately to divide the numbers written in scientific notation.
The Quotient of Powers Property
According to this property, when dividing two powers with the same base, we subtract the exponents.
\(a^m\div a^n=a^{m-n}\)
Example 4:
Find \(\frac{1\times 10^{-8}}{4\times 10^7}\)
Solution:
\(\frac{1\times 10^{-8}}{4\times 10^{7}}=\frac{1}{4}\times \frac{10^{-8}}{10^7}\) Divide 1 by 4.
\(=0.25\times 10^{-15}\) The quotient of powers property.
\(=(2.5\times 10^{-1})\times 10^{-15}\) Write 0.25 in scientific notation.
\(=2.5\times 10^{-16}\) The product of powers property.
Solved Examples on Scientific Notations
Example 1:
Calculate the sum: \((2.7\times 10^5)+(9.72\times 10^5)\).
Solution:
\(=(2.7+9.72)\times 10^5\) The distributive property.
\(=12.42\times 10^5\) Add.
\(=(1.242\times 10^1)\times 10^5\) Write 12.42 in scientific notation.
\(=1.242\times 10^6\) The product of powers property
Example 2:
Calculate the difference: \((5.6\times 10^{-3})-(2.3\times 10^{-4})\).
Solution:
Rewrite \(2.3\times 10^{-4}\) to have the same power of ten as \(5.6\times 10^{-3}\)
\(2.3\times 10^{-4}=2.3\times 10{-1}\times 10^{-3}\) Rewrite \(10^{-4}\) as \(10^{-1}\times 10^{-3}\).
\(=0.23\times 10^{-3}\) Rewrite \(2.3\times 10^{-1}\) as 0.23.
Subtract the factors.
\((5.6\times 10^{-3})-(0.23\times 10^{-3})\)
\(=(5.6-0.23)\times 10^{-3}\) The distributive property.
\(=5.37\times 10^{-3}\) Subtract.
Example 3:
Find \((2\times 10^{-5})\times (13\times 10^{-4})\)
Solution:
By the commutative property of multiplication,
\((2\times 10^{-5})\times (13\times 10^{-4})=2\times 13\times 10^{-5}\times 10^{-4}\)
\(=(2\times 13)\times (10^{-5}\times 10^{-4})\) The associative property of multiplication
\(=26\times 10^{-9}\) Simplify.
\(=(2.6\times 10^1)\times 10^{-9}\) Write 26 in scientific notation.
\(=2.6\times 10^{-8}\) The product of powers property
Example 4:
Find \(\frac{3\times 10^{-7}}{8\times 10^6}\)
Solution:
\(\frac {3\times 10^{-7}}{8\times 10^{6}}=\frac{3}{8}\times \frac{10^{-7}}{10^6}\)
\(=0.375\times \frac{10^{-7}}{10^6}\) Divide 3 by 8.
\(=0.375\times 10^{-13}\) The quotient of powers property.
\(=(3.75\times 10^{-1})\times 10^{-13}\) Write 0.375 in scientific notation.
\(=3.75\times 10^{-14}\) The product of powers property.
Scientific notation is an important topic in mathematics that must be thoroughly understood in order to develop the ability to easily represent numbers that have a lot of digits. These numbers, when written down as they are, would be cumbersome. Hence, knowledge of scientific notation is vital.
In scientific notation, “e” represents the exponent of 10.
For example, \(5.9\times 10^{-9}\) can be represented using “e” as 5.9e – 9.