**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 non-zero real number*a*and two integers*m*and*n*, (*a*)^{m}=^{n}*a*^{m}.^{n} - Product of Powers Property
**:**This property states that to multiply powers having the same base, add the exponents.

That is, for a real number non-zero*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 non-zero 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 non-zero 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 non-zero real numbers*a*and*b*and any integer*m*, .

**Examples of Power Properties**

- 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}= 2^{7}= 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^{4-3}= 5^{1}= 5. - Power of a Quotient Property: is the same as .

**Solved Example on Power Properties**

Evaluate:

Choices:

A. 823,543

B. 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:= 7^{5}= 16,807[Simplify.]

Step 3:So, .

**Related Terms for Power Properties**

- Power
- Product
- Quotient
- Factor
- Exponent