Raising a Product to a
Power
Consider how we would simplify a product raised to a positive power and a product
raised to a negative power using known rules.
In each of these cases the original exponent is applied to each factor of the product.
These examples illustrate the power of a product rule.
Power of a Product Rule
If a and b are nonzero real numbers and n is any integer, then
(ab)^{n} = a^{n} Â· b^{n}.
Example
Using the power of a product rule
Simplify. Assume the variables represent nonzero real numbers. Write the answers
with positive exponents only.
a) (3x)^{4}
b) (2x^{2})^{3}
c) (3x^{2}y^{3})^{2 }
Solution
a) (3x)^{4} 
= (3)^{4}x^{4} 
Power of a product rule 

= 81x^{4} 

b) (2x^{2})^{3} 
= (2)^{3}(x^{2})^{3} 
Power of a product rule 

= 8x^{6} 
Power of a power rule 
