# Powers of Integers

The power of a natural exponent of an integer is another integer. The absolute value of the result is the absolute value of the base multiplied by itself as specified by the exponent. The sign of the result can be determined by the following rule:

1. The powers of an even exponent are always positive.

2. The powers of an odd exponent have the same sign of the base.

### Properties

1. a0 = 1

2. a1 = a

3. Multiplication of powers with the same base:

It is another power with the same base and the exponent is the sum of the exponents.

am · a n = am+n

(−2)5 · (−2)2 = (−2)5+2 = (−2)7 = −128

4. Division of powers with the same base:

It is another power with the same base and the exponent is the difference of the exponents.

am : a n = am — n

(−2)5 : (−2)2 = (−2)5 — 2 = (−2)3 = −8

5. Power of a power:

It is another power with the same base and the exponent is the product of the exponents.

(am)n = am · n

[(−2)3]2 = (−2)6 = 64

6. Multiplication of powers with the same exponent:

It is another power with the same exponent, whose base is the product of the bases.

an · b n = (a · b) n

(−2)3 · (3)3 = (−6)3 = −216

7. Division of powers with the same exponent:

It is another power with the same exponent, whose base is the quotient of the bases.

an : b n = (a : b) n

(−6)3 : 33 = (−2)3 = −8