# Powers

### Powers with an Integer Exponent

### Powers with a Fractional Exponent

### Properties

1,** a ^{0} = 1**

2** a ^{1} = a**

3. **Multiplication of powers with the same base**:

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

**a ^{m } · a ^{n } = a^{m+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 between the exponents.

**a ^{m } : a ^{n } = a^{m — 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.

**(a ^{m})^{n }= a^{m · 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

**a ^{n } · 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.

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

(−6)^{3} : 3^{3 } = (−2)^{3} = −8