Theory:

Product law
The product law states that the exponents can be added when multiplying two powers with the same base.
 
\(a^{m} \times a^{n} = a^{m + n}\), where \(a \ne 0\) and \(a\), \(m\), \(n\) are integers.
Example:
1. \(3^4 \times 3^2\)
 
Here, the base \(3\) is the same for both the numbers. So, we can add the powers.
 
\(a^{m} \times a^{n} = a^{m + n}\)
 
\(3^4 \times 3^2 = 3^{4 + 2} = 3^{6}\)
 
 
2. \(5^{-4} \times 5^{-2}\)
 
Method I:
 
\(5^{-4} \times 5^{-2}\) \(=\) 154×152
 
\(=\) 154×52
 
\(=\) 154+2 \(=\) 156 \(=\) \(5^{-6}\)
 
Thus, \(5^{-4} \times 5^{-2}\) \(=\) \(5^{-6}\).
 
Method II:
 
 \(5^{-4} \times 5^{-2}\) \(=\) \(5^{(-4)+(-2)} = 5^{-6}\)
Quotient law
The quotient law states that we can divide two powers with the same base by subtracting the exponents.
 
aman=amn, where \(a \ne 0\) and \(a\), \(m\), \(n\) are integers.
Example:
1. 5452=542=52
 
 
2. 4642=462
 
\(=\) \((-4)^{6+2}\) \(=\) \((-4)^{8}\)
 
Therefore, 4642=48.
Power law
The power law states that when a number is raised to a power of another power, we need to multiply the powers or exponents.
 
\((a^m)^n = a^{mn}\), where \(a \ne 0\) and \(a\), \(m\), \(n\) are integers.
Example:
1. \((5^2)^3 = (5)^{2 \times 3} = 5^{6}\).
 
2. \([5^{(-2)}]^3 = 5^{(-2) \times 3} = (5)^{-6}\).
 
3. \([(-5)^{2}]^{3} = (-5)^{2 \times 3} = (-5)^{6}\).
 
4. \([(-5)^{2}]^{-3} = (-5)^{2 \times (-3)} = (-5)^{-6}\).