Positive Rational Exponents
When a rational number is used as an exponent, the denominator indicates â€œrootâ€
and the numerator indicates â€œpower.â€ For example, 8^{2/3} means (8^{1/3})^{2}. We take the
cube root of 8 to get 2, then square 2 to get 4. Thus
8^{2/3} = 4.
In general, a^{m/n} is the mth power of the nth root of a.
Rational Exponents
If m and n are positive integers, then
a^{m/n} = (a^{1/n})^{m},
provided that a^{1/n} is defined.
Example 1
Evaluating expressions with positive rational exponents
Evaluate each expression.
a) 16^{3/4}
b) 27^{4/3}
c) (8)^{2/3 }
d) 4^{3/2 }
Solution
a) To evaluate 16^{3/4}, take the fourth root of 16 to get 2 and then cube 2 to get 8.
These steps are written as follows:
16^{3/4} = (16^{1/4})^{3} = 2^{3} = 8
b) To evaluate 27^{4/3}, take the cube root of 27 to get 3, and then raise 3 to the fourth
power to get 81:
27^{4/3} = (27^{1/3})^{4} = 3^{4} = 81
c) (8)^{2/3} = [(8)^{1/3}]^{2} = (2)^{2} = 4
d) 4^{3/2} = (4^{1/2})^{3} = 2^{3} = 8
Caution
We can evaluate 4^{3/2} because the negative sign is saved
until last. But (4)^{3/2} is not a real number because (4)^{1/2} is not a real number.
By the definition of rational exponents we find the root and then the power. For
example,
8^{2/3} = (8^{1/3})^{2} = 2^{2} = 4.
However, we get the same result if we find the power and then the root:
8^{2/3} = (8^{2})^{1/3} = 64^{1/3} = 4
In general, a^{m/n} is also the nth root of the mth power of a.
Evaluating a^{m/n} in Either Order
If m and n are positive integers, then
a^{m/n} = (a^{1/n})^{m} = (a^{m})^{1/n},
provided that a^{1/n} is defined.
The fact that the power can be found before the root will be used later in this
chapter. It is not useful in simply evaluating expressions because if the power is
evaluated first, we might have to find the root of a very large number. For example,
finding the power first in Example 1(b), we get
27^{4/3} = (27^{4})^{1/3} = 531,441^{1/3}
Calculator Closeup
It does not matter whether
you use a fraction or a decimal
as the exponent as long as the
decimal is the exact value of
the fraction. You will get the same result
whether you find the root first
or the power first.
Helpful hint
Note that in a^{m/n} we do not require
m/n to be reduced. As
long as the nth root of a is
real, then the value of a^{m/n} is
the same whether or not m/n is in its lowest terms.
