# Powers of Natural Numbers

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

**5 · 5 · 5 · 5 = 5 ^{4 }**

#### 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. **a ^{0} = 1 **

2. **a ^{1} = 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.

**a ^{m} · a ^{n } = a^{m + n}**

2^{5 } · 2^{2 } = 2^{5 + 2 } = 2^{7}

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.

**a ^{m } : a ^{n } = a^{m − n}**

2^{5 } : 2^{2 } = 2^{5 − 2 }= 2^{3}

5. **Power of a power: **

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

**(a ^{m})^{n} = a^{m · n }**

(2^{5})^{3} = 2^{15}

6. **Multiplication of powers with the same exponent**:

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

**a ^{n } · b ^{n } = (a · b) ^{n}**

2^{3 } · 4^{3 } = 8^{3}

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.

**a ^{n } : b^{n } = (a : b)^{n}**

6^{3 } : 3^{3 } = 2^{3}

**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 · 10 ^{3} + 6 · 10^{2} + 5 · 10^{1} + 8 **

Negative Exponents, Fractional Exponent, Exponents Worksheets.