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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}.