Roots

For even values of n , the expression is defined to be the positive nth root of a or the principal nth root of a. For example, denotes the positive second root, or square root, of a , while is the positive fourth root of a. When n is odd, there is only one n th root, which has the same sign as a. For example, , the cube root of a, has the same sign as a. By definition, if then . On a calculator, a number is raised to a power using a key labeled For example, to take the fourth root of 6 on a TI-83 calculator, enter , to get the result 1.56508458.

EXAMPLE 1

Calculations with Exponents

Rational Exponents

In the following definition, the domain of an exponent is extended to include all rational numbers.

DEFINITION OF

For all real numbers a for which the indicated roots exist, and for any rational number m/n

EXAMPLE 2

Calculations with Exponents

NOTE could also be evaluated as but this is more difficult to perform without a calculator because it involves squaring 27, and then taking thecube root of this large number. On the other hand, when we evaluate it as we know that the cube root of 27 is 3 without using a calculator, and squaring 3 is easy.

All the properties for integer exponents given in this section also apply to any rational exponent on a nonnegative real-number base.

EXAMPLE 3

Simplifying Exponential Expressions

In calculus, it is often necessary to factor expressions involving fractional exponents.

EXAMPLE 4

Simplifying Exponential Expressions

Factor out the smallest power of the variable, assuming all variables represent positive real numbers.

Solution

To check this result, multiply by

Solution

The smallest exponent here is 3. Since 3 is a common numerical factor, factor out

Check by multiplying. The factored form can be written without negative exponents as

.

Solution

There is a common factor of 2. Also, have a common factor. Always factor out the quantity to the smallest exponent. Here -1/2 < 1/2 so the common factor is and the factored form is