Power Properties
Definition of Power Properties
Power of a Power Property: This property states that the power of a power can be found by multiplying the exponents.
That is, for a nonzero real number a and two integers m and n, (a^{m})^{n} = a^{mn}.
Product of Powers Property: This property states that to multiply powers having the same base, add the exponents.
That is, for a real number nonzero a and two integers m and n, a^{m} � a^{n} = a^{m+n}.
Quotient of Powers Property: This property states that to divide powers having the same base, subtract the exponents.
That is, for a nonzero real number a and two integers m and n, .
Power of a Product Property: This property states that the power of a product can be obtained by finding the powers of each factor and multiplying them.
That is, for any two nonzero real numbers a and b and any integer m, (ab)^{m} = a^{m} � b^{m}.
Power of a Quotient Property: This property states that the power of a quotient can be obtained by finding the powers of numerator and denominator and dividing them. That is, for any two nonzero real numbers a and b and any integer m,.
Video Examples: Power of a Power Property
Example of Power Properties

In the above figure, the letter R is on the top.
 Power of a Power Property: (2^{2})^{3} = 4^{3} = 64 is the same as 2^{2�3} = 2^{6} = 64.
 Product of Powers Property: 2^{2} � 2^{5} = 4 � 32 = 128 is the same as 2^{2+5} = 27 = 128.
 Power of a Product Property: (3 � 4)^{2} = 12^{2} = 144 is the same as 3^{2} � 4^{2} = 9 � 16 = 144.
 Quotient of Powers Property: is the same as 5^{43} = 51 = 5.
 Power of a Quotient Property: is the same as .
Solved Example on Power Properties
Ques: Evaluate:
Choices:
A. 823,543B. 16,807
C. 2,401
D. 117,649
Correct Answer: B
Solution:

Step 1: [To divide powers with same base, subtract their exponents.]
Step 2: = 75 = 16,807 [Simplify.]
Step 3: So, .