TUTORIALS:

Rational Exponents
Definition â€”
Rational Exponent With Numerator m
If a is a real number, m is a nonnegative integer, n is a positive
integer, and
is in lowest terms, then
If n is even, then am/n is a real number only when a
≥ 0.
Examples:
Note:
When m = 1, we have
That is
Example 1
Rewrite using a radical and evaluate: 32^{3/5 }
Solution
We will use three different methods to solve this problem.
Method 1 Use

32^{3/5 } 
Here, a = 32, m = 3, and n = 5. 

Find the prime factorization of 32.
32 = 2 Â· 2
Â· 2 Â· 2
Â· 2 = 2^{5}. 

Simplify the radical.
Evaluate 2^{3}. 
= (2)^{3} = 8 
Method 2 Use


Comparing a^{m/n} to 32^{3/5} we see that a
= 32,
m = 3, and n = 5. 

Use your calculator to find 32^{3}. 

Use your calculator to find

= 8 
Method 3 Use the Power of a Power Property
of Exponents, (a^{m})^{n} = a^{mn}.

32^{3/5 } 
Find the prime factorization of 32.
32 = 2 Â· 2
Â· 2 Â· 2
Â· 2 = 2^{5}. 
= (2^{5})^{3/5} 
Use the Power of a Power Property of Exponents.


Multiply the exponents and write the result
in lowest terms.
Evaluate 2^{3}. 
= 2^{3}
= 8 
In summary, all three methods lead to the same result: 32^{3/5 }
= 8
Note:
Method 2 typically means dealing with
larger numbers since we first evaluate the
power.
