Also, we will generalize this concept and introduce nth roots. The definition of the principal (non-negative) square root follows.

Simplify.

Not all square roots work out so nicely. If trying to simplify a square root with a radicand that is not a perfect square, we can find the exact answer using the following properties.

Here A and B are non-negative real numbers and B is not equal to zero. Approximations are made using a calculator.

Give the exact answer and approximate to the nearest hundredth.

This idea can be extended to any positive integer index n.

For 3rd roots, or cube roots, the question is “what raised to the 3rd power will produce the given number?" For example,

because

Simplify.

Remember that if the radicand of an odd root is negative then the result will be a negative real number. If the radicand of an even root is negative then the number is not real.

The numerator and denominator of these rational expressions contain fractions and look very intimidating. We will outline two methods for simplifying them.

Method 1: Obtain a common denominator for the numerator and denominator, multiply by the reciprocal of the denominator, then factor and cancel if possible.

Method 2: Multiply the numerator and denominator of the complex fraction by the LCD of all the simple fractions then factor and cancel if possible.

To illustrate what happened after we multiplied by the LCD we could add an extra step.

For the following solved problems, both methods are used. Choose whichever method feels most comfortable for you.