# Powers of Natural Numbers

A power is an abbreviated form of writing a multiplication formed by several equal factors.

5 · 5 · 5 · 5 = 54

#### Base

The base of a power is the number that multiplies by itself, in this case, 5.

#### Exponent

The exponent of a power indicates the number of times to multiply the base by itself, in this case, 4.

### Properties of the Powers of Natural Numbers

1. a0 = 1

2. a1 = a

3. Product 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

25 · 22 = 25 + 2 = 27

4. Division of powers with the same base:

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

am : a n = am − n

25 : 22 = 25 − 2 = 23

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

(25)3 = 215

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

23 · 43 = 83

7. Division of powers with the same exponent:

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

an : bn = (a : b)n

63 : 33 = 23

### Polynomial Decomposition of a Number

A natural number can be decomposed using powers with the base of 10.

The number 3,658 can be decomposed as follows:

3,658 = 3 · 103 + 6 · 102 + 5 · 101 + 8