an-m = an/am

an+m = an ∙ am

a-n = 1/an

anm = anam = (an)m = (am)n

anbn = (ab)n

n√a = a(1/n)

n√(a b) = n√a ∙ n√b

n√(a/b) = ( n√a) / ( n√b)

n√(am) = (am)(1/n) = a(m/n) = (n√a)m

(a/b)n = an / bn

a0 = 1

a1 = a

**Example 1:
**

*2*^{4}/2^{2} = (2*2*2*2 ) / (2*2) = 2^{4-2} = 2^{2}

**
**

**Example 2:
**

*2*^{3}2^{4} = 2^{3+4} = 2^{7}^{
}

**Example 3:
**

*2*^{4}3^{4} = (2*3)^{4} = 6^{4}^{
}

**
**

**Example 4:
**

*[3/2]*^{4} = 3^{4} / 2^{4}

**Example 5:
**

*3*^{-2} = 1 / 3^{2}

**Example 6:
**

*(3*^{2})^{4} = 3^{(2*4)} = 3^{8
}

__
__

**Exponential Powers Grow Expressions Rapidly
**

Exponential powers increase the value of an expression at an incredibly large rate. In order to see this, consider the following example:

2^{1} = 2

2^{2} = 4

2^{3} = 8

2^{4} = 16

2^{5} = 32

2^{6} = 64

2^{7} = 128

2^{8} = 256

2^{9} = 512

In each instance, the value is doubling, which makes sense since it is being multiplied by 2 another time for each increase in the value of the exponent.**
**

**
**

**Example 7:
**

A carpenter cuts a wooden box randomly. He finds that one of the pieces cut was square in shape and its side measure was 115 of a meter. Find the area of the square.

__
__

Area of a square = side x side = s^{2}

Substitute in place of s, (s= 115)

Area of a square = 1225

The area of the square is 1225m^{2}.

**Example 8**:

Find the value of m that makes the expression (5j^{m})^{3} = 125j^{9} valid.

__
__

125j^{9} can be written as (5j^{3} x 5j^{3} x 5j^{3}), which is the same as (5j^{3} x 5j^{3} x 5j^{3}) = (5j^{3})^{3}.

So, the value of m in the expression (5j^{m})^{3} is 3.