Radical Notation

If a is a real number, n is a positive integer, and is a real number, then

If a is a real number, n is a positive integer, and is a real number, then

Example

Evaluate each root: a) b) c) - d) e) -

f)

Solution:

a) = 3 since 3= 81

b) = -2 since (-2)= -8

c) - since

d) is not a real number. There is no real number that when raised to the 4th

power is equal to a negative number.

e) since 2= 64

f) since

Rules for Radicals

For all real numbers a and b, and positive integers m and n for which the indicated roots are real numbers,

A radical expression is simplified when the following conditions are satisfied.

1. All possible factors have been removed from the radicand.

2. There is no fraction in the radicand.

3. There are no radicals in the denominator.

4. The index of the radical is reduced.

Rationalizing the denominator

1. If the denominator is a monomial, multiply both the numerator and denominator by the same radical so that the resulting denominator is rational (contains no radical).

For example,

2. If the denominator is a binomial, multiply both the numerator and denominator by the conjugate of the denominator. For example:

Example

Simplify each radical expression. Assume that all variables represent positive real numbers.

a) b) c) d) e)

f) g) h) i)

Solution:

a)

b)

c)

d)

e)

f)

g)

h)

i)

Example

Rationalize the denominator of the radical expression. Assume that all variables represent nonnegative numbers and that no denominators are zero.

a) b)

Solution:

a)

b)