Properties of Exponents
The table below lists several properties of exponents. Here, m and n are
real numbers.
Name

Property 
Example 
Multiplication Property 
x^{m} Â· x^{m}
= x ^{m + n} 
5^{4} Â· 5^{2}
= x ^{4 + 2} = 5^{6} 
Division Property 


Power of a Power
Property 


Power of a Product
Property 
(xy)^{n} = x^{n} y^{n} 
(2x)^{3} = 2^{3} x^{3 }= 8x^{3} 
Power of a Quotient Property 


Zero Power Property 
x^{0} = 1, x ≠ 0 
17^{0} = 1 
Negative Exponent 


Example 1
Simplify: 2xy^{4}(3x^{5}y)(x^{3}y^{5})^{2}
We can drop parentheses around the
quantity (3x^{5}y) since it has an exponent
of 1. 
Solution 
2xy^{4}(3x^{5}y)(x^{3}y^{5})^{2} 
That is, (3x^{5}y) = (3x^{5}y)^{1}
= 3x^{5}y. 
Use the Power of a Product Property.

= 2xy^{4}(3x^{5}y)(x^{3Â·2}y^{5Â·2}) 

Simplify.

= 2xy^{4}(3x^{5}y)(x^{6}y^{10})
= 2xy^{4} Â· 3x^{5}y
Â· x^{6}y^{10} 
Recall: x = x^{1} and y = y^{1 } 
Multiply the constants, 2 and 3. 
= 6xy^{4} Â· x^{5}y
Â· x^{6}y^{10} 

Use the Multiplication Property of Exponents.

= 6x^{1+5+6 }y^{4+1+10} 
So, 6xy^{4} = 6x^{1}y^{4 }and x^{5}y
= x^{5}y^{1}. 
Simplify. 
= 6x^{12 }y^{15} 

So, the result is 6x^{12}y^{15}. 

