Theory:

Standard Notation
A number in standard notation is simply the number as we usually write it.
Example:
The number \(256\) in standard form.
 
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Scientific notation
A number is said to be in the scientific form when written as \(k \times 10^n\), where \(1 \le k < 10\) and \(n\) is an integer. Scientific notation allows us to express a very small or very large number in a compact form.
Example:
The number \(256\) in scientific form.
 
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Procedure to write the number in scientific notation:
Step 1: A number to be indicated as the product of a number (integer or decimal) and a power of \(10\).
 
Step 2: The number of steps moved is taken as the exponent of \(10\).
 
Step 3: Move the decimal point after the highest place value of the given number.
 
Step 4: Now, add a power of \(10\) that tells how many places you moved the decimal right or left.
 
Step 5: If you move the decimal place to the right, then the power will have a negative exponent. If you move the decimal places to the left, then the power will have a positive exponent.
Example:
1. Write the scientific form of the number \(5624.032\).
 
Move the decimal place to after \(5\) (left side).
 
Number of steps to be moved \(=\) \(3\)
 
\(5624.032\) \(=\) \(5.624032 \times 10^3\).
 
 
2. Write the scientific form of \((3.7 \times 10^{-5}) (3 \times 10^{-3})\).
 
\((3.7 \times 10^{-5}) (3 \times 10^{-3})\) \(=\) \((3.7 \times 3) \times (10^{-5-3})\)
 
\(=\) \(11.1 \times (10^{-8})\)
 
\(=\) \(1.11 \times (10^{-8 + 1})\)
 
Here, one decimal place moved to the left side, so \(1\) is added in the exponent of \(10\).
 
\(=\) \(1.11 \times (10^{-7})\)
 
Therefore, \((3.7 \times 10^{-5}) (3 \times 10^{-3})\) \(=\) \(1.11 \times 10^{-7}\).
 
 
3. Write in scientific form \((3.7 \times 10^{5}) (3 \times 10^{3})\).
 
\((3.7 \times 10^{5}) (3 \times 10^{3})\) \(=\) \((3.7 \times 3) \times (10^{5 + 3})\)
 
\(=\) \(11.1 \times (10^{8})\)
 
\(=\) \(1.11 \times (10^{8+1})\)
 
Here, one decimal place moved to the left side, so \(1\) is added in the exponent of \(10\).
 
\(=\) \(1.11 \times 10^{9}\)
 
Therefore, \((3.7 \times 10^{5}) (3 \times 10^{3})\) \(=\) \(1.11 \times 10^{9}\).
Important!
1. The positive exponents in scientific form indicates that it is a large number.
 
2. The negative exponents in scientific form indicates that it is a small number.