The laws of exponents **are the rules to follow to perform operations with powers. **The power of a number is the result of multiplying that number by itself more than once. The number is called the base, and the times it is multiplied is the exponent, which is placed small above and to the right of the base.

**a ^{n} = base ^{exponent}**

## 1) Power with zero exponent and nonzero base

Every number with exponent 0 (that is, raised to zero) is equal to 1.

**For instance:**

a ^{0} = 1

2 ^{0} = 1

15 ^{0} = 1

## 2) Power with exponent equal to one

Every number with exponent 1 is equal to itself. Examples of this would be the following:

a ^{1} = a

10 ^{1} = 10

15 ^{1} = 15

## 3) Product of powers of equal base

To multiply powers of the same base, the exponents are added, such as:

to ^{3} . a ^{5} = (a. a. a) (a. a. a. a. a) = a ^{3 + 5} = a ^{8}

**For instance:**

2 ^{3} . 2 ^{3} = 2 ^{3 + 3} = 2 ^{6} = 2. 2 . 2 . 2 . 2 . 2 = 64

to ^{15} . a ^{0} = a ^{15 + 0} = a ^{15}

4 ^{b} . 4 ^{c} = 4 ^{b + c}

## 4) Division of powers of equal base

To divide powers of the same base, the exponents are subtracted.

**For instance:**

a ^{10} ÷ a ^{3} = a ^{10 – 3} = a ^{7}

b ^{3} ÷ b ^{4} = b ^{3 – 4} = b ^{-1} = 1 / b

x ^{23} / x ^{13} = x ^{23 – 13} = x ^{10}

Every number with a negative exponent is equal to its inverse with a positive exponent, as shown below:

Another way to understand the division of powers is by eliminating common terms in the numerator and denominator, such as:

## 5) Law of uniformity

If the two members of an equality are raised to the same power, another equality results.

**For instance:**

a = 3

⇒ a ^{2} = 3 ^{2} ⇒ a ^{2} = 9

⇒ a ^{3} = 3 ^{3} ⇒ a ^{3} = 27

## 6) Power of a product

It is also known as the **distributive law of empowerment with respect to multiplication** . This law establishes that the multiplication ( *abc* ) raised to the *n* (nth power) is equal to each of the factors raised to that power and then multiplied.

**For instance:**

We can demonstrate this in the following way:

**For instance:**

(2 x 3) ^{3} = 2 ^{3} x 3 ^{3} = (2.2.2) (3.3.3) = 8 x 27 = 216

(3ab) ^{2} = 3 ^{2} . to ^{2} . b ^{2} = 9 a ^{2} b ^{2}

## 7) Power of a fraction

It is also known as the **distributive law of empowerment with respect to exact division** . To raise a fraction to a power, its numerator and denominator are raised to that power as follows:

**For instance:**

In the case of a mixed fraction, the number is transformed into a fraction:

## 8) Power of a power

If we multiply powers of equal base and equal exponent, we will have a power of another power:

to ^{m} . to ^{m} . a ^{m} multiplied *n* times = (a ^{m} ) ^{n} = a ^{m. n}

b ^{3} . b ^{3} . b ^{3} = (b ^{3} ) ^{3} = b ^{3.x 3} = b ^{9}

To solve the power of a power, we leave the same base and multiply the exponents:

(2 ^{4} ) ^{2} = 2 ^{4 x 2} = 2 ^{8} = 256

## 9) Law of monotony

When the two members of an inequality are greater than zero and rise to the same nonzero power, an inequality of the same sense results.

**For instance:**

5> 3

⇒ 5 ^{2} > 3 ^{2} ⇒ 25> 9

⇒ 5 ^{3} > 3 ^{3} ⇒ 125> 27

See also Hierarchy of operations .