Theory:

For any positive numbers \(a\) and \(b\), we have:
 
1. ab=a×b=a×b
 
2. ab=ab
Example:
1. Find the value of 81×36.
 
Solution:
 
We need to find the value of 81×36.
 
Apply the product rule of square root.
 
ab=a×b=a×b
 
81×36=81×36
 
=92×62
 
=9×6  (square and square root get cancelled.)
 
\(=\) \(54\)
 
Therefore, the value of81×36 \(=\) \(54\).
 
 
2. Simplify: 4875
 
Solution:
 
4875=16×325×3
 
Now, cancel the common factor \(3\).
 
4875=1625
 
Apply the quotient rule, ab=ab, \(b \ne 0\).
 
4875=1625
 
4875=45
 
Therefore, 45 is the simplified value of 4875.
 
 
3. Find the value of 10.24 by applying the quotient rule.
 
Solution:
We can write 10.24 \(=\) 1024100.
 
Now, apply the quotient rule ab=ab, \(b \ne 0\).
 
10.24 =1024100
 
=322102
 
=3210
 
\(=\) \(3.2\)
 
Therefore, 10.24 \(=\) \(3.2\).
Multiplying identical square root numbers
If we multiply the square root number by itself, we get the same number without the square root.
 
a×a=a
Example:
\(\sqrt{2} \times \sqrt{2} = 2\), \(\sqrt{3} \times \sqrt{3} = 3\)