5.1 Exponential Functions, 6.3 Exponential Functions

Exponents: Simplify

Example:  Rewrite the expression in the form 5^p where p is an algebraic expression.

25/root{3}{5}

Solution:

The original expression
25/root{3}{5}
Rewrite the numerator with a base of 5. 
25=5^2
25/root{3}{5}=5^2/root{3}{5}
 
Rewrite the radical as a fractional exponent.  In general root{n}{x^m}=x^{m/n}.  For this example root{3}{5}= 5^{1/3}
5^2/root{3}{5}=5^2/5^{1/3}
 
Use the quotient rule for exponents to subtract the exponents.
Quotient Rule for Exponents
b^m/b^n=b^{m-n}
5^2/5^{1/3}=5^{2-1/3}
 
Simplify by getting a common denominator and combining the fractions.
5^{2-1/3}=5^{6/3-1/3}=5^{5/3}